Analytic signal in many dimensions
Mikhail Tsitsvero Pierre Borgnat Paulo Gonçalves
eess.SP cs.IT math.CV math.IT math.SP


[
    Rodrigo Olea
    December 30, 2023
=====================




In this work we extend analytic signal theory to the multidimensional case when oscillations are observed in the d orthogonal directions. First it is shown how to obtain separate phase-shifted components and how to combine them into instantaneous amplitude and phases. Second, the proper hypercomplex analytic signal is defined as holomorphic hypercomplex function on the boundary of certain upper half-space. Next it is shown that correct phase-shifted components can be obtained by positive frequency restriction of hypercomplex Fourier transform. Necessary and sufficient conditions for analytic extension of the hypercomplex analytic signal into the upper hypercomplex half-space by means of holomorphic Fourier transform are given by the corresponding Paley-Wiener theorem. Moreover it is demonstrated that for d>2 there is no corresponding non-commutative hypercomplex Fourier transform (including Clifford and Cayley-Dickson based) that allows to recover phase-shifted components correctly. 




§ INTRODUCTION


[lraise=0.015, lines=3]Analytic signal is a holomorphic/analytic complex-valued function defined on the boundary of upper complex half-plane. The boundary of upper half-plane coincides with  and therefore analytic signal is given by the mapping f_a: →. The problem of extending a complex-valued function defined on the boundary of a complex region was originally posed by Riemann and later by Hilbert <cit.>. In <cit.> Hilbert introduced transformation that relates real and imaginary parts of analytic function on the boundary of an open disk/upper half-plane, that is now known as Hilbert transform, and is used to define analytic signal. As from the beginning of XX-th century the concepts of amplitude and phase of a complex function became ubiquitous in quantum mechanics. Additionally in 1946 Gabor <cit.> proposed to use analytic signal to encode instantaneous amplitude and phase for communication signals. Analytic signal representation of oscillation processes is peculiar because it allows to define the concepts of instantaneous amplitude, phase and frequency in unique and convenient way <cit.>. During the last few decades an interest emerged towards extension of analytic signal concept to multidimensionsional domains, mainly motivated by the problems coming from the fields ranging from image/video processing to multidimensional oscillating processes in physics like seismic, electromagnetic and gravitational waves. In other words by all the domains that focus on oscillating processess in more than one spatial/temporal direction. 

Typically one is not able to fully describe oscillating process in several dimensions by using just complex numbers due to insufficient degrees of freedom of complex numbers. In this case one hopes to rely on algebraic constructions that extend ordinary complex numbers in a convenient manner. Such constructions are usually referred to as hypercomplex numbers <cit.>. 









While complex numbers allow to relate oscillations of the form

    f(x) = a(x) cosϕ(x)

to the complex Euler form

    f_a(x) = a(x) e^i ϕ(x),

the goal of this paper is to relate oscillations of the form 

    f(x_1, …, x_d  ) = a(x_1, …, x_d) cosϕ_1 ( x_1 ) cosϕ_2 ( x_2 ) ·…·cosϕ_d ( x_d )

to the corresponding hypercomplex Euler form 

    f_S ( x_1, …, x_d ) = a_S ( x_1, …, x_d ) e^e_1 ϕ_1 ( x_1 )· e^e_1 ϕ_2 ( x_2 )·…· e^e_d ϕ_d ( x_d ).


 A number of works addressed various issues related to the proper choice of hypercomplex number system, definition of hypercomplex Fourier transform and partial Hilbert transforms for the sake of studying instantaneous amplitude and phases. Mainly these works were based on properties of various spaces such as ^d, quaternions, Clifford algebras and Cayley-Dickson constructions. In the following we list just some of the works dedicated to the studies of analytic signal in many dimensions. To the best of our knowledge, the first works on multidimensional analytic signal appeared in the early 1990-s including the work of Ell on hypercomplex transforms <cit.>, the work of Bülow <cit.> and the work of Felsberg and Sommer on monogenic signals <cit.>. Since then there was a huge number of works that studied various aspects of hypercomplex signals and their properties. Studies of Clifford-Fourier transform and transforms based on Caley-Dickson construction applied to the multidimensional analytic signal method include <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>. Partial Hilbert transforms were studied by <cit.>, <cit.>, <cit.> and others. Recent studies on Clifford-Fourier transform include <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and many others. Monogenic signal method was recently studied and reviewed in the works <cit.>, <cit.> and others. Unfortunately we cannot provide the complete list of references, however we hope that reader can find all the relevant references for example in the review works <cit.> and <cit.>. 


The hypercomplex analytic signal f_S is expected to extend all useful properties that we have in 1-D case. First of all one should be able to extract and generalize instantaneous amplitude and phases to d dimensions. Second, we expect that hypercomplex Fourier transform of analytic signal will be supported only over a positive quadrant of hypercomplex space. Third, conjugated parts of complex analytic signal are related by Hilbert transform, so we can expect that conjugated components in hypercomplex space should be related also by some combination of Hilbert transforms. And finally it should be possible to extend hypercomplex analytic signal to some uniquely defined holomorphic function inside a region of hypercomplex space. 

We address these issues in sequential order. First of all we start without anything that is hyper- or complex- by considering the Fourier integral formula and showing that Hilbert transform is related to the modified Fourier integral formula. This fact allows definition of instantaneous amplitude, phase and frequency without any reference to hypercomplex numbers and holomorphic functions. We proceed by generalizing the modified Fourier integral formula to several dimensions and defining the phase-shifted components that can be combined into instantaneous amplitude and phases. Second, we address the question of existence of holomorphic functions of several hypercomplex variables. It appears that commutative and associative hypercomplex algebra generated by the set of elliptic (e^2_i = -1) generators is a suitable space for hypercomplex analytic signal to live in. We refer to such a hypercomplex algebra as Scheffers space and denote it by _d. Hypercomplex analytic signal is defined as an extension of a hypercomlex holomorphic function of several hypercomplex variables to the boundary of certain upper half-space that is denoted by ^d_+. We then observe the validity of Cauchy integral formula for functions ^d_+ →_d, where the integral is calculated over tori-like hypersurfaces inside ^d_+ and deduce the corresponding partial Hilbert transforms that relate hypercomplex conjugated components. Finally it appears that hypercomplex Fourier transform of analytic signal with values in Scheffers space is supported only over non-negative frequencies, while one can use holomorphic Fourier transform for extension of analytic signal inside the upper half-space _+^d.

The paper is organised as follows. In Section <ref>, we present general theory of analytic signal, establish modified Fourier integral formula and relate it to the Hilbert transform. In Section <ref>, it is shown how to obtain phase-shifted components in several dimensions and combine them to define the instantaneous amplitude, phases and frequencies. Then, in Section <ref> we describe general theory of hypercomplex holomorphic functions of several hypercomplex variables and introduce Cauchy integral formula as well as Hilbert transform for polydisk and upper half-space. Next, in Section <ref>, we review the commutative hypercomplex Fourier transform and show that analytic signal can be reconstructed by positive frequency restriction of the hypercomplex Fourier transform of the original real-valued oscillating process. We conclude the section by proving Bedrosian's theorem for hypercomplex analytic signal. In Section <ref>, necessary and sufficient conditions for extension of analytic signal by means of holomorphic Fourier trasform are given by the Paley-Wiener theorem for functions with positively supported hypercomplex Fourier transform. In Section <ref> we prove that positive frequency restriction of the class of non-commutative hypercomplex Fourier transforms (including Clifford and Cayley-Dickson) does not provide correct phase-shifted components for d > 2. The paper is concluded by Section <ref> where we provide several illustrative examples of amplitude computation as well as observe limitations of the approach in the case when oscillations are not aligned with the chosen orthogonal directions of the ambient space.














§ IN ONE DIMENSION


  Let us assume that we have some oscillating function f: ℝ→ℝ and we want to obtain its envelope function, that “forgets” its local oscillatory behavior, and instantaneous phase, that shows how this oscillatory behavior evolves. In one dimension one gets analytic signal by combining the original signal with its phase-shifted version. The phase-shifted version is given by the Hilbert transform f̃ of the original function f
 
    f̃ = H [ f ] (x) = p.v.1/π∫_ℝf(y)/x-y  y =  lim_ε→ 01/π∫_|x-y|>εf(y)/x-y  y,   x ∈ℝ.


 The analytic signal f_a : ℝ→ℂ is obtained by combining f with f̃, i.e. f_a = f + i f̃, where i^2 = -1. This definition comes from the fact that analytic signal is extension of holomorphic (analytic) function in the upper half-plane ℂ_+ to its boundary ∂ℂ_+ = ℝ, i.e. f and f̃ are harmonic conjugates on the boundary ∂ℂ_+. Basic theory of Hilbert transform is concisely given in Appendix <ref>. We will need several standard definitions.
 
 The Fourier transform of a function f: → is defined as 
 
    f̂(ω) = F [ f ](ω) = ∫_ f(x) e^-i ω x  x, 
       f(x) = F^-1[ f̂] (x) = 1/2 π∫_f̂(ω) e^i ω x ω.



 The sign function is defined as 
 
    (x) = 
     -1    if  x < 0, 
    
     0    if  x = 0, 
    
     1    if  x > 0.

 
 It turns out that F[f_a](ω) is supported only on [0,∞) and therefore spectrum of f_a does not have negative frequency components which frequently are redundant for applications. This is due to the following relation.
  
 	Suppose f: → is continuous and continuously differentiable, then
 	
    F [ f̃] (ω) = ∫_-∞^∞ H[f](x)   e^-i ω x  x = -i sign(ω) F[f](ω)

 
 Combination of the function f and its Hilbert transform f̃ provides all the necessary information about the envelope function, instantaneous phase and frequency.
 
 	
 	 

 
 	
  * The absolute value of complex valued analytic signal is called instantaneous amplitude or envelope a(x)
 	
    a(x) = |f_a(x)| = √(f(x)^2+f̃(x)^2).

 	
  * The argument ϕ(x) of analytic signal f_a(x) is called instantaneous phase
 	
    ϕ(x) =  [ f_a ](x) = arctan( f̃(x)/f(x)).

 	
  * The instantaneous frequency ν(x) is defined as derivative of instantaneous phase
 	
    ν(x) = ϕ/ x.

 
 
 
 	
 	 Analytic signal f_a(x) could be defined in terms of Fourier transform, by discarding the negative frequency components 
 	 
    f̂_a(ω) = [ 1 +  (ω) ] f̂(ω).
 
 
 
 Quite complementary perspective on the relationship between f and f̃ may be obtained by observing that 
 
    H [ cos (ω x' + ϕ) ](x) = sin(ω x + ϕ).

 Therefore we observe how Hilbert transform phase-shifts the cosine by π/2 turning it into sine. Next we give definitions of Fourier sine and cosine transforms that will appear useful in defining the phase-shifted components. 
 
 The Fourier cosine and sine transforms of continuous and absolutely integrable f:→ are defined by
 
    F_c [f](ω) =  ∫_-∞^∞ f(x) cos(ω x)   x, 
    
     F_s [f](ω) =  ∫_-∞^∞ f(x) sin(ω x)   x.

 
 
 In the following we will rely on the Fourier integral formula that follows directly from the definition of Fourier transform.
 
 	Any continuous and absolutely integrable on its domain f: → may be represented as
 
    f(x)= 1/π∫_0^∞∫_-∞^∞ f(x') cos[ (x-x') ω]   x' ω.

 
 	From the definition of Fourier transform we can write the decomposition
 	
    f(x)= 1/2 π∫_-∞^∞∫_-∞^∞ f(x') e^ (x-x') ω  x' ω.

 	Taking the real part of both sides and observing that cosine is even in ω we arrive to the result.
 
 
 
 In the following theorem we deduce the modified Fourier integral formula and relate it to the Hilbert transform.
 
 	
 	Let f: → be continuous and absolutely integrable and let H[f] be the Hilbert transform of f. Then we have
 	
    H[f](x) =  1/π∫_0^∞∫_-∞^∞ f(x') sin[ (x-x') ω]   x'  ω.

 	
		Let us denote for convenience the left-hand side of (<ref>) by f̃ and right-hand side by f_1. In the proof we treat the generalized function δ(x) as an infinitely localized measure or a Gaussian function in the limit of zero variance. Under this assumption, we can write the following cumulative distribution function
		
    H(x) = ∫_-∞^x δ(s)   s,

 		which is the Heaviside step function H(ω) = 1/2(1+(ω)), and, in particular, we have
 		
    H(0)= ∫_-∞^0δ(s)   s = ∫_0^∞δ(s)   s = 1/2.

 		Then we write the Fourier transform of the right hand side of (<ref>) as
 		
    F [f_1] (ω')     =
     		1/π∫_-∞^∞( ∫_0^∞∫_-∞^∞ f(x') sin[ ω (x-x') ]   x' ω)  e^-i ω' x  x 
        = 1/2 π i∫_-∞^∞∫_0^∞∫_-∞^∞ f(x') [ e^i ω (x-x') - e^-i ω (x-x')] e^-i ω' x  x' ω x 
        = 1/2 π i∫_-∞^∞∫_0^∞ F[f](ω) e^i(ω-ω')x ω x - 1/2 π i∫_-∞^∞∫_0^∞ F[f](-ω) e^i(-ω-ω')x ω x  
        = -i ∫_0^∞ F[f](ω) δ(ω - ω')  ω + i ∫_0^∞ F[f](-ω) δ(-ω - ω')  ω.

 		In the derivations above we used the relation ∫_-∞^∞ e^i(ω-ω')x  x = 2 πδ(ω - ω'). 
 		Now we see, by taking into account (<ref>), that
 		
    ∫_0^∞ F[f](ω) δ(ω - ω')  ω = H(ω') F[f](ω'),  
       ∫_0^∞ F[f](-ω) δ(-ω - ω')  ω = H(-ω') F[f](ω').

 		Finally, from the last line in (<ref>) we observe that
 		
    F[f_1](ω) = 
     		-i F[f](ω),    if ω > 0, 
    
     		- i/2 F[f](ω) +  i/2 F[f](-ω),    if ω = 0, 
    
     		i F[f](ω),    if ω < 0,

 		which is exactly in accordance with (<ref>) and therefore f_1 = f̃.  		











 	
 

 Now by expanding sine of difference in the integrand of (<ref>), we can write
 
    f_1(x) =    1/π∫_0^∞∫_-∞^∞ f(x') [ sin( x ω) cos( x' ω) - cos(x ω) sin( x' ω) ]   x'  ω
    
     =    1/π∫_0^∞( ∫_-∞^∞ f(x') cos( x' ω)  x' ) sin( x ω)  ω
        - 1/π∫_0^∞( ∫_-∞^∞ f(x') sin( x' ω)  x' ) cos( x ω)  ω.

 For the sake of brevity we introduce the following notation
 
    α_0 = α_0 (x, ω) =cos( ω x ), 
       α_1 = α_1 (x, ω) = sin( ω x ), 
       α^0 = α^0 (ω) = ∫_-∞^∞ f(x') cos( x' ω)   x' 
       α^1 = α^1 (ω) = ∫_-∞^∞ f(x') sin( x' ω)   x', 
       α̂^0 = α̂^0 (x) = 1/2π∫_-∞^∞f̂(ω) cos( ω x )  ω, 
       α̂^1 = α̂^1 (x) = 1/2π∫_-∞^∞f̂(ω) sin( ω x )  ω.

 For the integration over ω of a pair of functions a(ω) and b(x,ω) we will write
 
    ⟨ a, b ⟩_+    = 1/π∫_0^∞ a(ω) b(x,ω)  ω, 
    ⟨ a, b ⟩   = 1/2π∫_-∞^∞ a(ω) b(x,ω)  ω.


 
 
 	
 	Equipped with the notation introduced in (<ref>) and (<ref>) and using Fourier integral formula (<ref>) we can expand f_0 = f(x) in the following form
 	
    f_0 = ⟨α^0, α_0 ⟩_+ + ⟨α^1, α_1 ⟩_+.

 	Similarly using the results of Theorem <ref> we can write for f_1 = f̃
 	
    f_1 = ⟨α^0, α_1 ⟩_+ - ⟨α^1, α_0 ⟩_+.

 
 Above observation shows that the original signal f is obtained by first projecting f onto sine and cosine harmonics and then by reconstructing using the same harmonics correspondingly. Hilbert transform f_1, or phase-shifted version of f, is obtained differently though. First we project initial function f on sine and cosine harmonics to obtain α^1 and α^0, however then for reconstruction the phase-shifted harmonics are used, i.e. α_0 and α_1.
 
 	The Fourier transform of a general complex-valued function f: →, with f(x) = f_0(x) + i f_1(x), may be written in terms of projection on phase-shifted harmonics as
 	
    f̂(ω) = F[f](ω) = α^0(ω) - i α^1(ω),

 	which follows if we apply Euler's formula e^ix = cos x + i sin x to the definition (<ref>). The inverse transform may be written as
 	
    F^-1 [ f̂ ](x) = α̂^0(x) + i α̂^1(x).

 	The original function is recovered by
 	
    f(x) = F^-1[ f̂] (x) = ⟨α^0, α_0 ⟩ + ⟨α^1, α_1 ⟩ + i ( ⟨α^0, α_1 ⟩ - ⟨α^1, α_0 ⟩).

 
 A variety of approaches for generalization of the analytic signal method to many dimensions has been employed. For example quaternionic-valued generalization of analytic signal has been realized <cit.> and works well in case d=2. Later on, in the Example <ref>, we show why and how the quaternionic based approach is consistent with presented theory. As a next step we are going to generalize formulas (<ref>) and (<ref>) to multidimensional setting, first, without relying on any additional hypercomplex structure, while later in Sections <ref> and <ref> we choose a convenient hypercomplex algebra and Fourier transform.
 
 


§ IN MANY DIMENSIONS


In several dimensions we are puzzled with the same questions on what are the adequate definitions of instantaneous amplitude and phase. Again we start by considering some function f : ℝ^d →ℝ having some local oscillatory behavior. We start by extending the Fourier integral formula to many dimensions.

For convenience and brevity we introduce the following notation. For a given binary {0,1}-vector j∈{0,1}^d we define the functions α_j: ^d ×^d → by

    α_j = α_j (x, ω) = ∏_l=1^dcos(ω_l x_l - j_l π/2).



 
	Any continuous and absolutely integrable over its domain f: → may be decomposed as
	
    f(x_1, …, x_d)=     1/π^d∫_0^∞∫_-∞^∞…∫_0^∞∫_-∞^∞ f(x'_1,x'_2,…,x'_d) cos[ (x_1 - x'_1) ω_1 ] ·…
       ·cos[ (x_d - x'_d) ω_d ]   x'_1 ω_1 … x'_d ω_d.

	or by employing the notation (<ref>),
	
    f(x) = 1/π^d∫_0^∞∫_-∞^∞ f(x') α_0(x-x', ω)  x'ω
 
	with 0∈{ 0,1 }^d being the all-zeros vector.
	
		Proof is the same as for 1-D case.
	
 
 Similarly to the result of Theorem <ref> next we define the phase-shifted copies of the oscillating function f.
 
 	The phase-shifted version f_j: →, in the direction j∈{0,1}^d, of the function f: → is given by
 	
    f_j(x) = 1/π^d∫_0^∞∫_-∞^∞ f(x') α_j(x-x', ω)  x'ω.

 
Next we give the definitions of instantaneous amplitude, phase and frequency. 
 
 	  
 

 	
 		
  * The square root of sum of all phase-shifted copies of signal is called instantaneous amplitude or envelope a(x)
 		
    a(x) =  √(∑_j∈{0,1}^d f_j (x)^2).

 		
  * The instantaneous phase ϕ_j(x) in the direction j∈{0,1}^d is defined by
 		
    ϕ_j(x) = arctan( f_j(x)/f(x)).

 		
  * The instantaneous frequency ν_j(x) in the direction j∈{0,1}^d is defined as partial derivatives of the corresponding instantaneous phase in the directions given by j
 		
    ν_j(x) = ∂^jϕ_j = ∂_1^j_1∘∂_2^j_2∘ ... ∘∂_d^j_d( ϕ_j) (x),

 		where we used multi-index notation for derivatives, ∂_i^1 = ∂/∂ x_i and ∂_i^0 = id. Note that no Einstein summation convention was employed in (<ref>).
 	
 
 We can generalize Theorem <ref> by relating phase-shifts f_j to the corresponding multidimensional Hilbert transforms. However first let us define Hilbert transform in several dimensions.
 
 	
 	The Hilbert transform in the direction j∈{0,1}^d is defined by
 	
    H_j[ f ] (x) =  p.v.1/π^|j|∫_^|j|f(y)/(x-y)^j y^j,

 	where |j| gives the number of 1-s in j, (x-y)^j = ∏_i=1^d (x_i - y_i)^j_i and y^j =  y_1^j_1 ...  y_d^j_d, i.e. integration is performed only over the variables indicated by the vector j.
  
 
 	
 	For a continuous f: → that is absolutely integrable in its domain, we have
 	
    f_j = H_j[ f ].

 	
 		proof follows the lines of the proof of Theorem <ref>.
 	
 

It is useful to express the phase-shifted functions f_j as a combination of harmonics α_j and corresponding projection coefficients α^i. We define projection coefficients similarly to (<ref>),
 
    α^j = α^j(ω) = ∫_-∞^∞ f(x) α_j( x, ω)  x, 
       α̂^j = α̂^j(x) = 1/(2 π)^d∫_-∞^∞f̂(ω) α_j( x, ω)  ω, 
       α̂_+^j = α̂_+^j(x) = 1/π^d∫_0^∞f̂(ω) α_j( x, ω)  ω

 and brackets, similarly to (<ref>), as
 
    ⟨ a, b ⟩_+ = ⟨ a, b ⟩_+ (x)= 1/π^d∫_0^∞ a(ω) b(x,ω)  ω, 
    ⟨ a, b ⟩ = ⟨ a, b ⟩ (x)= 1/(2π)^d∫_-∞^∞ a(ω) b(x,ω)  ω,

 for any two functions a(ω) = a(ω_1,…,ω_d) and b(x,ω) = b(x_1,…,x_d, ω_1, …, ω_d).
 It is useful to observe how phase-shifted functions f_j could be obtained from α^j. 
 
 	For a given function f: ^d →, one obtains phase-shifted copy f_j, in the direction j∈{0,1}^d, by 
 	
    f_j = ∑_i∈{0,1}^d (-1)^| ( i⊕j) ⊖i|⟨α^i⊕j, α_i⟩_+,

 	where ⊕ is a binary exclusive OR operation acting elementwise on its arguments and ⊖ is defined as following: 1 ⊖ 0 = 1 and the result is 0 otherwise.
 	
 		The proof follows immediately after we expand all the cosines and sines of difference in (<ref>) and make all the substitutions from (<ref>) and (<ref>).
 	
 


Short illustration may be helpful to understand the rule (<ref>). Let us suppose that we have j = (1,0,1,1,0) and i = (1,1,0,0,0). In this case i⊕j = (1 ⊕ 1, 0 ⊕ 1, 1 ⊕ 0, 1 ⊕ 0, 0 ⊕ 0) = (0,1,1,1,0). Then we have ( i⊕j) ⊖i = (0 ⊖ 1, 1 ⊖ 1, 1 ⊖ 0, 1 ⊖ 0, 0 ⊖ 0) = (0,0,1,1,0) therefore we will have “+” sign in front of ⟨α^01110, α_11000⟩_+ for f_10110. 









	Before moving to the theory of hypercomplex holomorphic functions and analytic signals, it will be useful to note that from the definition (<ref>) there will be in total 2^d different components f_j. It says us that most likely the dimensionality of hypercomplex algebra in which analytic signal should take its values should be also 2^d.


 

§ HOLOMORPHIC FUNCTIONS OF SEVERAL VARIABLES

 
 

 §.§ Commutative hypercomplex algebra

 
 The development of hypercomplex algebraic systems started with the works of Gauss, Hamilton <cit.>, Clifford <cit.>, Cockle <cit.> and many others. Generally a hypercomplex variable w is given by the linear combination of n hypercomplex units { e_i }
 
    w = ∑_i=0^n w_i e_i,

 with coefficients w_i ∈. Product rule for the units e_i is given by the structure coefficients γ^s_ij∈
 
    e_i e_j = ∑_s=0^n γ^s_ij e_s.

 Only algebras with unital element or module are of interest to us, i.e. those having element ϵ = 1 such that ϵ x = x ϵ = x. The unital element have expansion
 
    ϵ = ∑_i=0^n ϵ_i e_i.

 For simplicity we will always assume that the unital element ϵ = e_0 = 1.
 
 The units e_i,  i ≠ 0, could be subdivided into three groups <cit.>.
 
 	
  * e_i is elliptic unit  if e_i^2 = -e_0
 	
  * e_i is parabolic unit  if e_i^2 = 0
 	
  * e_i is hyperbolic unit  if e_i^2 = e_0.
 
 
 
 	General properties of some abstract hypercomplex algebra  that is defined by its structure constants γ^s_ij may be quite complicated. We will focus mainly on the commutative and associative algebras constructed from the set of elliptic-type generators. For example suppose we have two elliptic generators e_1^2 = -1 and e_2 = -1, then one can construct the simplest associative and commutative elliptic algebra as _2 = span_{ e_0 ≡ 1,e_1,e_2, e_3 ≡ e_1 e_2 }. The algebra have 2^2 = 4 units. For _2 we have the following multiplication table
 	

 	
 		e_0     e_1     e_2     e_3 
 
 		e_1     -e_0     e_3      -e_2 
 
 		e_2     e_3      -e_0     -e_1  
 
 		e_3     -e_2      -e_1      e_0 
 
 		 
  	tableMultiplication table for _2 algebra
 

 
 
 One could assemble various types of hypercomplex algebras depending on the problem. All of the above unit systems have their own applications. For example, numbers with elliptic unit relate group of rotations and translations of 2-dimensional Euclidean space to the complex numbers along with their central role in harmonic analysis. Algebra with parabolic units may represent Galileo's transformations, while algebra with hyperbolic units could be used to represent Lorentz group in special relativity <cit.>. In this work we mainly focus on the concepts of instantaneous amplitude, phase and frequency of some oscillating process. Elliptic units are of great interest to those studying oscillating phenomena due to the famous Euler's formula. We briefly remind how it comes. Taylor series of exponential is given by
 
    e^z = 1 + z/1! + z^2/2! + z^3/3! + ⋯ = ∑_n=0^∞z^n/n!.

 We could have various Euler's formulae. In elliptic case when z = e_i x, e_i^2 = -1, we can write from the above Taylor's expansion
 
    e^e_i x = cos (x) + e_i sin (x).

 When e_i^2 = 0 we have
 
    e^e_i x = 1 + e_i x.

 And finally when e_i^2 = 1 we get
 
    e^e_i x = cosh (x) + e_i sinh (x).

 It is due to the relation (<ref>) that one relies on the complex exponentials for analysis of oscillating processes. One may well expect that to obtain amplitude and phase information, one will need a Fourier transform based on the algebra containing some elliptic numbers.
  
 It was a question whether the chosen hypercomplex algebra for multidimensional analytic signal should be commutative, anticommutative, associative or neither. Whether it is allowed to have zero divisors or not. In Section <ref> we show that commutative and associative algebra not only suffices but also is essentially a necessary condition to define Fourier transform coherently with (<ref>). Based on these general considerations we define the simplest associative and commutative algebra for a set of d elliptic units. We call it elliptic Scheffers[Due to the work <cit.> of G. W. Scheffers (1866-1945)] algebra and denote it by _d.
 
 
 	The elliptic Scheffers algebra _d over a field  is an algebra of dimension 2^d with unit ϵ = e_0 = 1 and generators { e_1, …, e_d} satisfying the conditions e^2_i = -1,  e_i e_j = e_j e_i,  i,j = 1, …, d. The basis of the algebra _d consists of the elements of the form e_0 = 1,  e_β = e_β_1 e_β_2… e_β_d,  β = ( β_1, … , β_s ),  β_1 < β_2 < … < β_s,  1 ≤ s ≤ d. Each element w ∈_d has the form
 	
    w = ∑_β = 0^2^d - 1 w_β e_β = w_0 ϵ + ∑_s = 1^d ∑_β_1 < … < β_s w_β e_β,   w_β∈.

 
 If otherwise not stated explicitly we will consider mainly algebras over .
 To illustrate the definition above we can write any w ∈_2 in the form
 
    w = w_0 + w_1 e_1 + w_2 e_2 + w_12 e_1 e_2.

 
 	We will use three different notations for indices of hypercomplex units and their coefficients. The dimension of Scheffers algebra _d is 2^d therefore there always will be 2^d indices. However one can choose how to label them. First notation is given by natural numbering of hypercomplex units. For example for _2 we will have the following generators { e_0 ≡ 1, e_1, e_2, e_3 ≡ e_1 e_2 }. Second notation is given by the set of indices in subscript, i.e. for _2 we have the generators { e_0 ≡ 1, e_1, e_2, e_1,2}. Third notation uses binary representation j∈{0,1}^d. For _2 example generators will be labeled as { e_00≡ 1, e_10, e_01, e_11}. To map between naturally ordered and binary representations we use index function (·), for example in case of _2 we have (11) = {1,2} = 3 as well as we have (3) = 11 ∈{ 0,1}^2 or (1,2) = 3 ∈ [0,1,...,2^d-1]. Even though the function (·) is overloaded, in practice it is clear how to apply it.
 
 
 
 	
 The space _d is a Banach space with norm of w ∈_d defined by
 
    | w |__d = √(∑_β =0^2^d-1 | w_β |^2),  w_β∈,

 while sometimes we will write just | w | instead of | w |__d.
 
 One can check that _d is a unital commutative ring. The main difference of the elliptic Scheffers algebra from the algebra of complex numbers is that factor law does not hold in general, i.e. if we have vanishing product of some non zero a,b ∈_d, it does not necessarily follow that either a or b vanish. In case ab=0,  a,b ≠ 0,  a and b are called zero divisors. Zero divisors are not invertible. However what is of actual importance for us is that _d has subspaces spanned by elements {e_0, e_i}, each of which has the structure of the field of complex numbers and therefore factor law holds inside these subspaces. We will need this result to define properly the Cauchy integral formula later <cit.>.
 
 
 	The space (i),  i = 1,...,d, is defined by 
 	
    (i) = { a + be_i | a,b ∈}.

 
 Real and imaginary parts of an element ζ = a + b e_i are defined in an obvious way by (ζ) = a and (ζ) = b, while norm of ζ is given by | ζ |_(i) = √(a^2 + b^2). 
 Similarly to the complex analysis of several variables we define open disk, polydisk and upper/lower half-planes.
 
 	The unit disk (i) ⊂(i) of radius 1 centered at zero is defined as
 	
    (i) = { x ∈(i): |x| < 1 }.

 	Sometimes we will write (w_i, r) for a disk of radius r centered at w_i ∈(i). 
 
 
 
 	The upper half-plane _+(i) is given by
 	
    _+(i) = { a + be_i | a,b ∈; b>0 },

 	while the lower half-plane will be denoted by _-(i)
 	
    _-(i) = { a + be_i | a,b ∈; b<0 }.

 
 Finally we combine these spaces to construct the domain for hypercomplex functions of several variables.
 
 	
 	The total Scheffers space ^d is a direct sum of d subalgebras (i)
 	
    ^d = ⊕_i=1^d (i).

 
  
 	
 	The upper Scheffers space _+^d is a direct sum of d subalgebras _+(i)
 	
    _+^d = ⊕_i=1^d _+(i).

 


	The mixed Scheffers space, denoted by _+ j, is the direct sum of upper spaces marked by 1-s in a vector j∈{0,1}^d and lower spaces marked by 0-s
	
    _+ j = ( ⊕_i: j_i = 1_+ (i) ) ⊕( ⊕_i: j_i = 0_- (i) ).



  
 	
 	The space ^d is a Banach space with norm of a vector v∈^d, v = ( v_1, …, v_d ), each v_i ∈(i), given by
 	
    | v |_^d = √(∑_i | v_i |_(i)^2),  v_i ∈(i).

 	Sometimes we will write for simplicity | v | instead of | v |_^d.
 

  
 	For j∈{0,1}^d we define j-polydisk _j⊂^d as a product of |j| disks (i) 
 	
    _j = ∏_i ∈(j)(i)

 	and we will write for d-polydisk ^d = ∏_i = 1^d (i) when we take product of all d disks. Sometimes we will use polydisk of radius r centered at vector w∈^d which is denoted by _j(w,r) = ∏_i ∈(j)(w_i, r).
 
 
 
 	The boundary of ^d_+ coincides with ^d, i.e.
 	
    ∂_+^d = ^d_+∖^d_+ ≃^d.

 
 While each (i) is equivalent to the complex plane , the combination (i) ⊕(j) is not equivalent to ^2 because imaginary units have different labels – in a sense (i) ⊕(j) contains more information than ^2.
 	 
 

 §.§ Holomorphic functions of hypercomplex variable

 
The scope of this section is to define hypercomplex holomorphic/analytic functions of type f:^d →_d. Indeed we are interested in rather restricted case of mappings between hypercomplex spaces which finally will be consistent with our definition of analytic signal as a function on the boundary of the polydisc in ^d in full correspondence with 1-dimensional analytic signal. There are some similarities and differences with theory of functions of several complex variables. We refer the reader to the book <cit.> for details on the function theory of several complex variables and to the concise review <cit.> to refresh in memory basic facts on complex analysis. We start by providing several equivalent definitions of a holomorphic function of a complex variable by following Krantz <cit.>. Then we define holomorphic function of several hypercomplex variables in a pretty similar fashion. Starting from the very basics will be instructive for the hypercomplex case.


The derivative of a complex-valued function f: → is defined as a limit

    ∂ f/∂ z( z_0 ) = lim_z → z_0f(z) - f(z_0)/z-z_0.

If the limit exists one says that f is complex differentiable at point z_0.



	If f: Ω→, defined on some open subset Ω⊆, is complex differentiable at every point z_0 ∈Ω, then f is called holomorphic on Ω.


Complex differentiability means that the derivative at a point does not depend on the way sequence approaches the point. If one writes two limits, one approaching in the direction parallel to the real axis and one parallel to the imaginary axis, then after equating the corresponding real and imaginary terms one obtains the Cauchy-Riemann equations and therefore equivalent definition of a holomorphic function.


	A function f: Ω→, explicitly written as a combination of two real-valued functions u and v as f(x+iy) = u(x,y) + i v(x,y), is called holomorphic in some open domain Ω⊆ if u and v satisfy the Cauchy-Riemann equations
	
    ∂ u/∂ x = ∂ v/∂ y, 
       ∂ u/∂ y = -∂ v/∂ x.



A holomorphic function of a complex variable z ∈ can be compactly introduced using the derivatives with respect to z and conjugated variable z̅

    ∂/∂ z = 1/2( ∂/∂ x - i ∂/∂ y), 
    ∂/∂z̅ = 1/2( ∂/∂ x + i ∂/∂ y).



	A function f: Ω→ is called holomorphic in some open domain Ω∈ if for each z ∈Ω
	
    ∂ f/∂z̅ = 0,

	which means that holomorphic function is a proper function of a single complex variable z and not of the conjugated variable z̅.

 Contour integration in the complex space is yet another viewpoint on the holomorphic functions. Complex integration plays the central role in our work due to our interest in Hilbert transform that is defined as a limiting case of contour integral over the boundary of an open disk (see Appendix <ref>).


	A continuous function f: Ω→ is called holomorphic if for each ω∈Ω there is an r = r(w)>0 such that (w,r) ⊆Ω and
	
    f(z) = 1/2 π i∮_|ζ - w| = rf(ζ)/ζ - z ζ

	for all z ∈(w,r).



	
	Structure of  as a field is important for the definition of the integral in (<ref>).  Only because every element of a field has a multiplicative inverse, we can write the integration of the kernel 1/z. However in imaginary situation when the contour of integration passes through some points where (ζ-z) is not invertible, we will be unable to write the Cauchy formula (<ref>).


Finally we can justify the word “analytic” in the title of the paper by giving the following definition. Note that analytic and holomorphic means essentially the same.

	A function f: Ω→ is called analytic (holomorphic) in some open domain Ω∈ if for each z_0 ∈Ω there is an r = r(z_0) >0 such that (z_0, r) ⊆Ω and f can be written as an absolutely and uniformly convergent power series
	
    f(z) = ∑_k a_k ( z - z_0 )^k

	for all z ∈( z_0,r ).


Having introduced basic notions of holomorphic functions now we are ready to review the holomorphic function theory in the hypercomplex space. Theory of holomorphic functions was generalized to the functions of commutative hypercomplex variables by G.W. Scheffers in 1893 <cit.>, then his work was extended in 1928 by P.W. Ketchum <cit.> and later by V.S. Vladimirov and I.V. Volovich <cit.>, <cit.> to the theory of superdifferentiable functions in superspace with commuting and anti-commuting parts. One important distinction to previous works is that we present different Cauchy formula from one given by Ketchum and Vladimirov. Their approach is based on complexification (i.e. w_β∈) of the underlying hypercomplex algebra, while we are working in the hypercomplex algebra over reals. We briefly describe their formula in Appendix <ref>.

Let us recall, having in mind Remarks <ref> and <ref>, the definition of continuity for a function f: A →_d defined on some open subset A ⊆^d. We say the limit of f as ζ approaches ζ_0 equals a, denoted by lim_ζ→ζ_0 = a if for all ε > 0 there exists a δ > 0 such that if ζ∈ A and | ζ - ζ_0 | < δ then | f ( ζ) - a | < ε. Then f is said to be continuous at ζ_0 if lim_ζ→ζ_0 = f(ζ_0). If f is continuous for every point ζ_0 ∈ A then f is said to be continuous on A.

First by following <cit.> we give definition of a holomorphic function in some general hypercomplex space Γ that is defined by its structure constants γ^s_ik, see (<ref>). This definition generalizes ordinary Cauchy-Riemann equations and follows from the same argument that hypercomplex derivative at a point should be independent on the way one approaches the point in a hypercomplex space. Finally we will restrict attention to the functions ^d →_d and will see that we are consistent with our previous derivations.

	A function f: Γ→Γ,  w ↦ f(w)	is holomorphic in an open domain Ω⊆Γ if for each point w ∈Ω it satisfies the generalized Cauchy-Riemann-Scheffers equations
	
    ∂ f_s/∂ w_k = ∑_i=0^nγ^s_ik∂ f_i/∂ w_0,  s,k = 0, …, n.


General equations could be simplified if we consider the functions that have only subsets of Γ = _d as their domain, in our case we consider functions f: (i) →_d that map plane (i) in _d to the total space _d. If we restrict our domain only to the plane (i) and insert values of structure constants for _d we obtain simple Cauchy-Riemann equations (see for example eqq. (<ref>) and (<ref>) for 2-D case). We will arrive to the explicit equations in a while. Mappings from restricted domains, i.e. in our case (i) ⊆_d, are described in details in <cit.>.


	To avoid confusion the comment on the relationship between the spaces (i), ^d and _d will be useful. While we can embed trivially (i) ↪^d as well as (i) ↪_d, one may tend to think that it will also be natural to embed ^d ↪_d. Last embedding always exists because dimensionality of _d is 2^d and is non smaller that the dimensionality of ^d which is 2d. However this embedding will not be trivial. For example consider an element x∈^d ⊆^d. In _d there is only one unital element, while in ^d there are d of them, because ^d = ⊕_i=1^d (i), therefore it is impossible to “match“ the unital elements of _d and ^d in a trivial way. Therefore it is better to think about ^d and _d as being two different spaces that however “share” planes (i).


From now on we will study the functions f: ^d →_d. A function f could be explicitly expressed as a function of several hypercomplex variables f(z_1, ..., z_d) ≡ f(x_1 + e_1 y_1, ..., x_d + e_d y_d) with each z_k ∈(k). 


A function f: ^d →_d is called hypercomplex differentiable in the variable z_k if the following limit exists

    ∂ f/∂ z_k (z_1, ..., z_0, ..., z_d) = lim_z → z_0f(z_1,...,z, ..., z_d)-f(z_1,...,z_0, ..., z_d)/z - z_0.




	If f: Ω→_d, defined on some open subset Ω⊆^d, is hypercomplex differentiable in each variable separately at every point z_0 ∈Ω, then f is called holomorphic on Ω.


One obtains the generalized Cauchy-Riemann equations using the same argument as in complex analysis. We give an example for a function f: ^2 →_2, which is easy to generalize for any d>2.

	Let us consider a function f: ^2 →_2. It depends on 4 real variables (x_1,y_1,x_2,y_2) because f(z_1, z_2) ≡ f(x_1+e_1 y_1, x_2 + e_2 y_2). Moreover in _2 we can expand the value of f componentwise
	
    f ( z_1, z_2 ) ≡ f ( x_1 + e_1 y_1, x_2 + e_2 y_2 ) =     f_0 ( x_1 + e_1 y_1,  x_2 + e_2 y_2 ) 
       + e_1 f_1 (x_1 + e_1 y_1,  x_2 + e_2 y_2) 
       + e_2 f_2 (x_1 + e_1 y_1,  x_2 + e_2 y_2) 
       + e_1 e_2 f_12 (x_1 + e_1 y_1,  x_2 + e_2 y_2).

	First we calculate the derivative in the “real” direction of the plane (1) as
	
    lim_Δ x → 0f(x_1 + Δ x + e_1 y_1, z_2)-f(z_1, z_2)/Δ x = 
    	∂ f_0/∂ x_1 + e_1 ∂ f_1/∂ x_1 + e_2 ∂ f_2/∂ x_1 + e_1 e_2 ∂ f_12/∂ x_1.

	Then calculate the derivative in the orthogonal direction e_1, note that 1/e_1 = e_1/e^2_1 = -e_1,
	
    lim_Δ y → 0f(x_1 +  e_1 y_1 + e_1 Δ y, z_2)-f(z_1, z_2)/ e_1 Δ y = 
    	-e_1 ∂ f_0/∂ y_1 + ∂ f_1/∂ y_1 - e_1 e_2 ∂ f_2/∂ y_1 + e_2 ∂ f_12/∂ y_1.


Equating the corresponding components gives us the Cauchy-Riemann equations for the variable z_1

    ∂ f_0/∂ x_1 = ∂ f_1/∂ y_1, 
       ∂ f_1/∂ x_1 = -∂ f_0/∂ y_1,
       ∂ f_2/∂ x_1 = ∂ f_12/∂ y_1,
       ∂ f_12/∂ x_1 = -∂ f_2/∂ y_1.

Similarly we get 4 equations for z_2

    ∂ f_0/∂ x_2 = ∂ f_1/∂ y_2, 
       ∂ f_1/∂ x_2 = -∂ f_0/∂ y_2,
       ∂ f_2/∂ x_2 = ∂ f_12/∂ y_2,
       ∂ f_12/∂ x_2 = -∂ f_2/∂ y_2.

These 8 equations provide necessary and sufficient conditions for a function f(z_1, z_2) to be holomorphic. It comes immediately after one observes that all three available directions e_0, e_1, e_2 are mutually orthogonal in the Euclidean representation of _2 and by linearity of the derivative operator.

Let us now express the conditions on holomorphic function using the derivative with respect to conjugated variable, where the conjugation of z_j = a + e_j b is defined by z̅_j = a - e_j b, j = 1, ...,d. Derivative operators are defined similar to (<ref>) by


    ∂/∂ z_j = 1/2( ∂/∂ x_j - e_j ∂/∂ y_j), 
    ∂/∂z̅_j = 1/2( ∂/∂ x_j + e_j ∂/∂ y_j).



	A function f: Ω→_d, defined on some open subset Ω⊆^d, is  holomorphic on Ω if
	
    ∂ f/∂z̅_j = 0,    for  j=1,...,d,

	at every point z_0 ∈Ω.

Again we give an example for f: ^2 →_2.

	The derivative with respect to z̅_1 of f: ^2 →_2 is given by
	
    ∂ f/∂z̅_1 = 1/2( ∂ f/∂ x_1 + e_1 ∂ f/∂ y_1) =    1/2( ∂ f_0/∂ x_1 + e_1 ∂ f_1/∂ x_1 +  e_2 ∂ f_2/∂ x_1 +  e_1 e_2 ∂ f_12/∂ x_1. 
       . + e_1 ∂ f_0/∂ y_1 -  ∂ f_1/∂ y_1 + e_1 e_2 ∂ f_2/∂ y_1 - e_2 ∂ f_12/∂ y_1).

	Equating the components in front of each e_i to zero we get equations (<ref>).


We are ready to introduce the Cauchy integral formula for the functions f: ^d →_d. The concept of Riemann contour integration as well as Cauchy's integral theorem are well defined in the unital real, commutative and associative algebras <cit.>. Some care will be taken to define Cauchy integral formula because some elements of _d are not invertible. To introduce the concept of integral in the space _d it is natural to assume (cf. (<ref>)) that

    w = ∑_β = 0^2^d-1 e_β w_β.

Therefore we understand the integral ∫ f(w)   w of _d-valued function f(w) of the variable w ∈_d as integral of differential form ∑_β f(w) e_β w_β along a curve in ^2^d. As it was pointed out in the Remark <ref>, presence of the inverse function in Cauchy formula implies that inverse of the function (z-ζ) does not pass through zero divisors in _d. It is true that _d contains some zero divisors. For example (e_1-e_2)(e_1+e_2) = 0. However if we restrict our attention to the functions (i) →_d then it is easy to see that each nonzero z ∈(i),  i = 1, ... ,d is invertible simply because each (i) is a field. Cauchy formula for f: (i) →_d for a disk (i) ⊂(i) will have the form

    1/2 π e_i∫_∂(i)f(ζ)/ζ - z ζ_i = 
    f(z)    if  z ∈(i), 
    
    0    if  z ∈(i) ∖(i).



Therefore we can also construct the multidimensional Cauchy integral formula for holomorphic functions f: ^d →_d without any problems. General Cauchy formula for a simple open polydisk _j⊂^d is thus given by

    1/(2 π)^|j| e_j∫_∂_jf ( ζ_j)/( ζ - z)^j ζ^j = 
    
    f(z)    if  z ∈_j, 
    
    0    if  z ∈^d ∖_j.

This motivates the following equivalent definition of a holomorphic function.



	Let function f: Ω→_d, defined on some open subset  Ω⊆^d, be continuous in each variable separately and locally bounded. The function f is said to be holomorphic in Ω if for each w∈Ω there is an r = r(w)>0 such that ^d (w,r) ⊆Ω and
	
    f(z_1, ..., z_d) = 1/( 2 π)^d e_1 ·…· e_d∮_|ζ_1 - w_1| = r…∮_|ζ_d - w_d| = rf ( ζ_1, …, ζ_d )/( ζ_1 - z_1 )·…·( ζ_d - z_d ) ζ_1 …ζ_d

	for all z∈^d(w,r).


Analytic hypercomplex function therefore will be defined as following.


	A function f: Ω→_d is called analytic (holomorphic) in some open domain Ω⊆^d if for each w∈Ω there is an r = r(w) >0 such that ^d (w, r) ⊆Ω and f can be written as an absolutely and uniformly convergent power series for all z∈^d(w,r)
	
    f(z) = ∑_i_1, …, i_d = 0^∞ a_i_1, …, i_d( z_1 - w_1 )^i_1·…·( z_d - w_d )^i_d

	with coefficients
	
    a_i_1, ..., i_d = 1/i_1! ·…· i_d!( ∂/∂ z_1)^i_1…( ∂/∂ z_d)^i_d f(z).



In case of a single complex variable Hilbert transform relates the conjugated parts of a holomorphic function on the boundary of a unit disk. Unit disk then can be mapped to the upper complex half-plane and we obtain the usual definition of the Hilbert transform on the real line. In case of polydisk in a hypercomplex space ^d we proceed similarly. First we define Hilbert transform on the boundary of unit polydisk and then consider all biholomorphic mappings from the polydisk to the upper space ^d_+.


	Hilbert transform H_j[ f ] of a function f: ∂_j→ is given by
	
    H_j[ f ] (  e^e_1 θ_1, ..., e^e_d θ_d) = 1/( 4 π)^|j|∫_-π^π f (e^e_i t_i | i ∈j) ∏_i ∈j( θ_i - t_i/2)  t^j,

	where integration of f is only over the variables z_i ≡ e^e_i t_i indicated by the binary vector j∈{0,1}^d.

To construct the holomorphic hypercomplex function from its real part, which is defined on the boundary of polydisk, we can find the corresponding hypercomplex conjugated components by f_j = H_j [f] and sum the up. The relationship with one-variable Hilbert transform is the following. In complex analysis each holomorphic function on the boundary of the unit disk has the form f_a = f + i H[f], i.e. Hilbert transform relates two real-valued functions. In case of several hypercomplex variables the situation is slightly different. The _d-valued function may be written for each j=1,…,d in the form f = f_1 + e_j f_2, however now f_1 and f_2 are not real-valued. Functions f_1 and f_2 have values lying in the {e_β | βj }. Therefore Hilbert transform in the variable z_j relates hypercomplex conjugates within the plane (j) by the expression f_2 = H_j [f_1].

Next we simply map the boundary of a polydisk to the boundary of Scheffers upper space. All biholomorphic mappings <cit.> from the unit polydisk to the upper Scheffers space, i.e. ^d →_+^d, have the form
	
    (w_1, ..., w_d ) ↦( a̅_1 w_1 - e^e_1 θ_1 a_1/w_1 - e^e_1 θ_1, …, a̅_d w_d - e^e_d θ_d a_d/w_d - e^e_d θ_d),  where  a_i ∈(i),  θ_i ∈.

By picking up one such mapping we easily define the Hilbert transform H_j on the boundary of ^d_+ and get (<ref>), similarly as it is described in Appendix <ref>. Hypercomplex analytic signal is defined to be any holomorphic function f: _+^d →_d on the boundary ∂_+^d ≃^d.

	The Scheffers hypercomplex analytic signal f_S : ^d →_d is defined on the boundary of upper Scheffers space ∂_+^d ≃^d from its real part f: ^d → by
	
    f_S ( x) = ∑_j∈{0,1}^d  e_j H_j[ f ] ( x).



In one varialbe complex analysis the Riemann mapping theorem states that for any simply connected open subset U ⊆ there exists biholomorphic mapping of U to the open unit disk in . However in case of several complex variables d ≥ 2 this result does not hold anymore <cit.>. Poincaré proved that in any dimension d ≥ 2 in case of d complex variables the ball is not biholomorphic to the polydisk. Even though the proof of Poincaré theorem in case of mappings ^d →_d is out of scope of this paper this conjecture has important consequences. We rely mainly on the d-polydisk ^d ⊂^d as a domain of the hypercomplex holomorphic function. If there is no biholomorphic mapping from polydisk in ^d to the ball in ^d then on their boundaries holomorphic functions will be quite different as well, i.e. it will be impossible to map holomorphic function defined as a limit on the boundary of polydisk to the holomorphic function defined on the boundary of a ball. There is one simple argument on why probably there is no biholomorphic mapping from boundary of open polydisk to the boundary of a ball. The boundary of polydisk is given by torus, while the boundary of a ball is given by hypersphere. Torus and sphere are not homeomorphic for d ≥ 2. Therefore it is hard to expect existence of a biholomorphic mapping between the two topologically different domains.

Analytic signal was defined as extension of some holomorphic function inside polydisk to its boundary that is torus. Extension of the mapping (<ref>) to the boundary relates points on torus T^d with points on ^d. There are other mappings that one can use to relate points on a compact shape with the points of ^d. On the other hand, shape that one chooses to represent ^d could also be used for convenient parametrization of analytic signal. For example, in one variable complex analysis, phase of analytic signal is given by the angle on the boundary of unit disk. Here we advocate the use of torus T^d as a natural domain not only to define analytic signal but also to parametrize analytic signal's phase. 

Suppose instead we defined analytic signal on a unit sphere in ^d. In this case if there is no biholomorphic mapping between ball and polydisk in ^d the definition of analytic signal as a limiting case of holomorphic function on the boundary of the ball will differ from the polydisk case. Very probable that we will not be able to employ partial Hilbert transforms H_j as simple relationships between conjugated components because there are no “selected” directions on the sphere. On the other hand, parametrization of analytic signal by a point on a sphere in ^d could provide an alternative definition for the phase of a signal in ^d. Sphere and torus both locally look similar to ^d. However torus is given by the product T^d = S^1 ×…× S^1 and ^d is given by the product ^d = ^1 ×…×^1, therefore there is a natural way to assign circle to each direction in ^d. In contrast hypersphere is a simple object that cannot be decomposed. Later in Observation <ref> we use parametrization on torus to describe certain class of analytic signals.


 

§ COMMUTATIVE HYPERCOMPLEX FOURIER TRANSFORM

 

 In this section we will define the Fourier transform that naturally arises for functions f: ^d →_d. We start by assuming that we are working in some appropriately defined Schwartz space 𝒮( ^d,_d ) of rapidly decreasing _d-valued functions <cit.>. Fourier transform F: 𝒮→𝒮 is an automorphism on 𝒮. After defining the Fourier transform we show that analytic signal is supported only on positive quadrant of frequency space, the property that is frequently desired in applications. In other words we are able to recover the phase-shifted functions f_j by restricting spectrum of the Fourier transform of a real-valued function to the positive quadrant in frequency space. After establishing basic facts about Fourier transform we proceed to the Bedrosian's theorem that allows one to easily construct the hypercomplex analytic signal in the Euler form from the real-valued oscillating function. 
 
 
 	The space 𝒮_d ( ^d,_d ) of rapidly decreasing functions f: ^d →_d is 
 	
    𝒮( ℝ^d, _d ) = { f ∈ C^∞(ℝ^d, _d ) :  f_α,β < ∞  ∀α, β∈ℕ^d },

 	where α, β are multi-indices, C^∞( ^d, _d ) is the set of smooth functions from ^d to _d, and
 	
    f_α,β = sup_x∈ℝ^d| x^α∂^β f(x) |.

 	To put it simply, Schwartz class contains all smooth functions for which all the derivatives go to zero at infinity faster than any polynomial with inverse powers.
 

 We define hypercomplex Fourier transform in terms of phase-shifted harmonics. Although the definition is a bit lengthy, we still think that this form allows one to better focus on the harmonic ”pieces“ of Fourier transform as well as to see how hypercomplex numbers are working and where the possible non-commutativity of various hypercomplex algebras could cause troubles.
 
 	The Fourier transform f̂ = F [ f ], with f̂: ^d →_d, of a function f: ^d →_d is defined by
 	
    f̂(ω) =     α^0 … 0 (ω) - ∑_i=1^d e_i α^0 … 1(i) … 0 (ω) 
       + ∑_i<j e_i e_j α^0 … 1(i,j) … 0(ω) - ∑_i<j<k e_i e_j e_k α^0 … 1(i,j,k) … 0(ω) + …,

 	where 0 … 1(i,j,k) … 0 means that there are 1-s on the i-th, j-th and k-th positions of binary string. The inverse Fourier transform is given by
 	
    f(x) =     α̂^0 … 0 (x) + ∑_i=1^d e_i α̂^0 … 1(i) … 0 (x) 
       + ∑_i<j e_i e_j α̂^0 … 1(i,j) … 0 (x) + ∑_i<j<k e_i e_j e_k α̂^0 … 1(i,j,k) … 0 (x) + ….

 
 This definition is equivalent (by applying Euler's formula) to the canonical form of hypercomplex Fourier transform
 
    f̂(ω_1,…,ω_d) = ∫_-∞^∞…∫_-∞^∞ f(x_1,x_2,…,x_d) e^ -e_1 ω_1 x_1·…· e^ -e_d ω_d x_d  x_1 … x_d, 
    
     f(x_1,x_2,…,x_d) = 1/( 2π)^d∫_-∞^∞…∫_-∞^∞f̂(ω_1,…,ω_d) e^e_1 ω_1 x_1·…· e^e_d ω_d x_d  x_1 … x_d.

 The inverse Fourier transform (<ref>) may be rewritten using (<ref>) as
 
    f(x) = ⟨f̂(ω), α_0…0⟩ + ∑_i e_i ⟨f̂(ω), α_0 … 1(i) … 0⟩ + ∑_i<j e_i e_j ⟨f̂(ω), α_0 … 1(i,j) … 0⟩ + ….

 Next we observe that phase-shifted functions f_j are easily recovered from the restriction of Fourier transform to only positive frequencies.
 
 	The function f_h : ^d →_d defined by the real-valued function f:^d → as 
 	
    f_h(x) = ⟨f̂(ω), α_0…0⟩_+ + ∑_i e_i ⟨f̂(ω), α_0 … 1(i) … 0⟩_+ + ∑_i<j e_i e_j ⟨f̂(ω), α_0 … 1(i,j) … 0⟩_+ + …

 	has as components the corresponding phase-shifted functions f_j, i.e.
 	
    f_h(x) =     f(x) + ∑_i e_i f_0 … 1(i) … 0 (x)
       + ∑_i<j e_i e_j f_0 … 1(i,j) … 0 (x) + ∑_i<j<k e_i e_j e_k f_0 … 1(i,j,k) … 0(x) + …,

 	and therefore f_h ≡ f_S.
 	
 		For a proof we simply put each term from the sum in (<ref>) inside (<ref>) and observe that components in (<ref>) are defined by (<ref>).
 	
 
 
 	The Fourier transform of f_S is given by
 	
    f̂_S(ω) = ∏_i=1^d[ 1 +  (ω_i) ] f̂ (ω_1,…,ω_d).

 
 
 In the following we proceed to the Bedrosian's theorem telling us that Hilbert transform of the product of a low-pass and a high-pass functions with non-overlapping spectra is given by the product of the low-pass function by the Hilbert transform of the high-pass function. We rely on the original work <cit.>. In the following (j) = {i| j_i=1 }, the set of positions of 1-s in the vector j∈{0,1}^d and (j) = {i| j_i=0 } is the complementary set.
 
 	
 	For a product of exponentials for a given j∈{0,1}^d 
 	
    h(x) = ∏_i ∈(j) e^e_i ω_i x_i,

 	we have the identity
 	
    H_j[ h ] (x) = ∏_i ∈ (j) e_i (ω_i)   e^e_i ω_i x_i.

 	
 		The result follows directly from the 1-dimensional case by succesive application of Hilbert transform.
 	
 
 
 	Let f,g: ^d →_d and let us given some j∈{0,1}^d. Suppose that for each k ∈(j) we have corresponding a_k>0 such that
 	
    f̂_k ( ω_k; x_1,…, x_k-1, x_k+1, … x_d ) = ∫_ f ( x_1, …, x_k, …, x_d )    e^-e_k ω_k x_k  x_k = 0  
         for all  |ω_k| > a_k

 	and
 	
    ĝ_k ( ω_k; x_1,…, x_k-1, x_k+1, … x_d ) = ∫_ g ( x_1, …, x_k, …, x_d )    e^-e_k ω_k x_k  x_k = 0  
        for all  |ω_k| ≤ a_k.

 	Then we have the identity
 	
    H_j[ f · g ] = f · H_j[ g ] .

 	
 		First we give the proof for 1-dimensional case and then extend it to our case. 
 		We can write the product of two functions a(x) and b(x) in Fourier domain as
 		
    a(x) b(x) = 1/(2 π)^2∫_∫_â(u) b̂(v) e^i(u+v)x  u  v

 		and
 		
    H [a · b] (x) = 1/(2 π)^2∫_∫_â(u) b̂(v) i (u+v) e^i(u+v)x  u  v.

 		If we assume that for some a>0 we have â(ω) = 0 for |ω| > a and b̂ (ω) = 0 for |ω| ≤ a, then the product â(ω) b̂(ω) will be non-vanishing only on two semi-infinite stripes on the plane { (u,v): |u|<a, |v|>a }. So for this integration region the value of the integral (<ref>) will not change if we replace
 		
    (u+v) → (v).

 		Then we will have
 		
    H [a · b] (x)    = 1/(2 π)^2∫_∫_â(u) b̂(v) i (v) e^i(u+v)x  u  v 
        = a(x) 1/2π∫_b̂(v) i  (v) e^ivx  v.

 		But we know from Lemma <ref> that
 		
    H [b](x) = 1/2 π∫_b̂ (v) H [ e^ivx'] ( x )   v = 1/2 π∫_b̂ (v) i (v) e^i v x  v.

 		So finally we have 
 		
    H[a · b] = a · H[b].

 		To prove the final result (<ref>), we succesively apply 1-dimensional steps taken in (<ref>), (<ref>) and the result of Lemma <ref>
 		
    H_j [f · g] (x) =     1/(2 π)^2 |j|∫_^|j|∫_^|j|f̂_j( u_(j); x_(j)) ĝ_j( v_(j); x_(j)) 
       ·∏_i ∈(j) e_i (u_i + v_i) e^e_i (u_i + v_i) x_i u^jv^j,

 		where we denoted by f̂_j( u_(j); x_(j)) and ĝ_j( v_(j); x_(j)) the Fourier transforms of f and g over the directions indicated by j. Thus f̂_j( u_(j); x_(j)) has as its arguments frequency variables indexed by (j) and the remaining spatial variables indexed by the complementary set (j). Finally by repeating the steps taken in (<ref>), (<ref>) we conclude that H_j [f · g]  = f · H_j [g].
 	
 
 
 
 	
 Suppose we have two real-valued functions A, B: ^d → that satisfy conditions (<ref>) and (<ref>) respectively for j = 1. Additionally suppose that B(x) = α_0(x,ω_0) for some choice of frequencies ω_0. Then we can write the analytic signal f_S of the product function C(x) = A(x) · B(x) in compact form
 
    C_S(x) = A(x) e^e_1 ω_1 x_1·…· e^e_d ω_d x_d.

 
 If we suppose more generally that B(x) is of the form
 
    B(x)= ∏_l=1^dcos( ϕ_l(x) )

 and satisfies condition (<ref>), we can write its analytic signal C_S(x) then as
 
    C_S(x) = A(x) e^e_1 ϕ_1(x)·…· e^e_d ϕ_d(x).

 For a 1-dimensional complex-valued function we interpret its multiplication by a complex number as rotation and scaling in the complex plane. Here we have similar geometric interpretation of narrowband hypercomplex analytic signal having the form (<ref>). First of all, each exponential e^e_i ϕ_i represent phase rotations in the corresponding hypercomplex subspace (i) by the angle ϕ_i. Therefore for C_S(x) in (<ref>) we have a combination of d separate rotations, i.e. the overall rotation will correspond to a point on the product of d circles S^1 ×…× S^1. The product of circles is a d-dimensional torus T^d (see also discussion in the end of Section <ref>). The total phase of some hypercomplex analytic signal (<ref>) therefore corresponds to some continuous map on torus γ: ^d → T^d. The amplitude function A(x) is attached to the points on the torus T^d. One last observation is that even though we know that the target domain _d of analytic signal is not a division ring (e.g. e_1 + e_2 is not invertible), we can claim that the value of narrow-band analytic signal of the form (<ref>) is always invertible thus leaving aside all the troubles concerning multiplicative inverse.




§ ANALYTIC EXTENSION INTO UPPER SPACE ^D_+



One can extend hypercomplex analytic signal f_a : ^d →_d to the holomorphic function in the upper hypercomplex space ^d_+ by using Poisson kernel (see Appendix <ref>) or by using holomorphic Fourier transform. The inverse holomorphic Fourier transform is given simply by extending the domain of usual inverse Fourier transform (<ref>) to _+^d by

    f(ζ_1, ζ_2, ... , ζ_d) = F^-1[ f̂] = 1/(2π)^d∫_-∞^∞…∫_-∞^∞f̂(ω_1,…,ω_d) e^e_1 ω_1 ζ_1· ... · e^e_d ω_d ζ_d  x_1 … x_d,

where each ζ_i ∈_+(i).

Next we wish to understand under what conditions the above integral is well behaved. In case the integral in (<ref>) is a nice integral, we could use for example Leibniz's rule to check whether generalized Cauchy-Riemann conditions (<ref>) are satisfied. We could fall into troubles in some cases, for example, when f̂ is supported on the whole ^d. It will make the modulus of exponentials e^e_i ω_i ζ_i grow exponentially fast with ω_i → - ∞, thus making the integral undefined. Fortunately we already know from the previous section that frequency support of our analytic signal f̂_S is [0, ∞ )^d. 

The necessary and sufficient conditions for extension of a function into some (hyper-)complex domain (not necessarily upper space) are given by various Paley-Wiener theorems <cit.>. In this work we are mainly interested in the extension of analytic signal into upper space ^d_+. We will focus on this case only, while other Paley-Wiener type theorems may be constructed similarly to those given in Chapter 7.2 of <cit.>.

We will need several technical definitions. First of all let us define the space of square integrable functions L^2 and Hardy space H^2 of integrable holomorphic functions as following.


	Let f: ^d →_d. We say f ∈ L^2 ( ^d, _d ) if
	
    ∫_-∞^∞| f(x_1, ..., x_d) |^2   x_1 ...  x_d < ∞,

	where norm | · | in the space _d is given by (<ref>). We say f ∈ L^2 ( ^d_+, _d ) when f is supported on [ 0, ∞)^d.



	
	Let f: _+^d →_d be holomorphic in ^d_+. We say that f ∈ H^2 ( ^d_+, _d ) if

	
    sup_y_i > 0 
     i = 1,...,d ∫_-∞^∞| f(x_1 + e_1 y_1, ..., x_d + e_d y_d) |^2   x_1 ...  x_d < ∞.


In general case, Hardy spaces H^p are defined separately for open disk and upper half-plane  (see Chapters 13.3 and 13.4 of <cit.> and <cit.>). For an arbitrary 1 < p ≤∞, Möbius transformation is not an isomorphism between Hardy spaces on disk and upper half-plane. Situation is different for p=2, where Möbius transformation of the form (<ref>) is an isometric isomorphism between Hardy spaces for upper space ^d_+ and polydisk ^d. Morevoer we can assume that H^2 is sufficiently large class of functions because for the case of one complex variable Lemma 13.3.2 in <cit.> tells us that if 0 < p_1 < p_2 < ∞, then H^p_2( , ) ⊊ H^p_1( , ). After these observations we let us work in space H^2 ( ^d_+, _d ). Finally we proceed to the central theorem that allows extension of hypercomplex analytic signal into upper space ^d_+ by means of holomorphic Fourier transform.


	
	Let f̂: ^d →_d with f̂∈ L^2 ( ^d_+ , _d ). Then f (ζ) = F^-1[ f̂ (ω) ] (ζ) is in H^2 ( ^d_+, _d ).
	
	Furthermore the usual inverse Fourier transform f(x_1,… ,x_d) of f̂(ω_1, …,ω_d) is the limit of f(ζ_1,…, ζ_d) ≡ f(x_1 + e_1 y_1,…, x_d + e_d y_d) as each y_i → 0^+ for i = 1,…,d in the following sense
	
    lim_(y_1,…,y_d) → (0^+,…,0^+)∫_-∞^∞| f(x_1 + e_1 y_1,…, x_d + e_d y_d) - f(x_1, …, x_d) |^2   x_1 … x_d = 0.

	
	Conversely, suppose f ∈  H^2 ( ^d_+, _d ) then there exist g ∈ L^2 ( ^d_+, _d ) such that g is an inverse holomorphic Fourier transform of f, i.e. f = F^-1[g].
	
		Proof is essentially the same as in case of one complex variable. Discussion may be found in <cit.>, see Theorem 7.2.4, or for example see Theorem 3.2 in <cit.>.
	


In one-variable complex analysis the holomorphic Fourier transform is an operator that is isometric isomorphism due to the Plancherel theorem. If we denote by _+ and _- the upper and lower complex half-planes respectively, Paley-Wiener theorem provides us with decomposition of the space of squarely integrable functions L^2 ( , ) = H^2 ( _+, ) ⊕ H^2 ( _-, ). Similar result could easily be obtained for functions ^d →_d, however as we could expect now we will have all possible 2^d factors of mixed spaces.

	We have the following decomposition
	
    L^2 ( ^d, _d ) = ⊕_j∈{0,1}^d_+ j,

	where spaces _+ j were defined in (<ref>). This is a concise version of Paley-Wiener theorem <ref>.

 


§ WHY NON-COMMUTATIVE FOURIER TRANSFORMS DOES NOT FIT FOR D>2?


There were propositions to apply Fourier transforms based on non-commutative algebras, like Clifford and Cayley-Dickson construction, to study instantaneous amplitude, phase and frequency. The aim of this section is to show that there is no non-commutative Fourier transform in the canonical form (<ref>), which lead us to the phase-shifted functions f_j after restriction of the support of Fourier transform only to the positive quadrant. It is interesting to note that Bülow and Sommer in the early work <cit.> mention slightly the commutative hypercomplex numbers and associated Fourier transform, however later studies were mainly focused on the non-commutative hypercomplex systems.


We make all the derivations for the case of Clifford algebra-valued Fourier transform and then find that result is also valid for any non-commutative algebra. First we make a brief introduction to the Clifford algebras. A Clifford algebra is a unital associative algebra that is generated by a vector space V over some field K, where V is equipped with a quadratic form Q: V → K. If the dimension of V over K is d and {e_1, …, e_d } is an orthogonal basis of ( V,Q ), then 𝒞l(V,Q) is a free vector space with a basis

    { e_i_1 e_i_2… e_i_k |  1 ≤ i_1 < i_2< … < i_k ≤ d  and  0 ≤ k ≤ d . }.

Element e_0 is defined as the multiplicative identity element. Due to the fact that V is equipped with quadratic form we have an orthogonal basis

    ( e_i, e_j ) = 0  for  i ≠ j,  and ( e_i, e_i ) = Q ( e_i ),

where ( ·, ·) denotes the symmetric bilinear form associated to Q. The fundamental Clifford identity implies that Clifford algebra is anticommutative

    e_i e_j = - e_j e_i  for  i ≠ j,  and ( e_i, e_i ) = Q ( e_i ),

which makes multiplication of the elements of Clifford algebra quite simple, we can put elements of a Clifford algebra in standard order simply by a number of swaps of neighbouring elements.
Every nondegenerate form Q can be written in standard diagonal form:

    Q(v) = v_1^2 + … + v_p^2 - v_p+1^2 - … - v_p+q^2,

with d = p+q. The pair of integers ( p,q ) is called the signature of the quadratic form. The corresponding Clifford algebra is then denoted as 𝒞l_p,q(). For our purposes of analytic signal construction (see discussion on elliptic units in Section <ref>) we will consider just Clifford algebras 𝒞l_0,q() over the field of real numbers , with the quadratic form 

    Q(e_i) = -1  for  1 ≤ i ≤ d.


Each element x ∈𝒞l_0,d( ) of Clifford algebra may be thought as a linear combination of its basis elements

    x = x_0 1 + ∑_i x_i e_i + ∑_i<j x_ij e_i e_j + ∑_i<j<k x_ijk e_i e_j e_k + …

with real coefficients and for which we also have identities e_i e_j = - e_j e_i  for  i ≠ j,  and  e_i^2 = -1.

First for our proof it will be handy to define the set-valued function Π that returns all permuted products of its arguments. For example for 3 arguments it acts as

    Π( a,b,c ) = { a · b · c,  a · c · b,  b · a · c,  c · a · b,  b · c · a,  c · b · a }.

For d arguments Π acts analogously. A particular Clifford-Fourier transform is then defined as

    f̂^𝒞l (ω_1,…,ω_d) = ∫_-∞^∞…∫_-∞^∞ρ( f(x_1,x_2,…,x_d), e^ -e_1 ω_1 x_1, …, e^ -e_d ω_d x_d)    x_1 … x_d,

where 
    ρ( f(x_1,x_2,…,x_d), e^ -e_1 ω_1 x_1, …, e^ -e_d ω_d x_d) ∈Π( f(x_1,x_2,…,x_d), e^ -e_1 ω_1 x_1, …, e^ -e_d ω_d x_d).


A Clifford-Fourier transform (<ref>) has inverse transform given by

    f ( x_1,…,x_d ) = 1/( 2 π)^d∫_-∞^∞…∫_-∞^∞ρ'( f̂^𝒞l(ω_1,ω_2,...,ω_d), e^ e_1 ω_1 x_1, …, e^ e_d ω_d x_d)    x_1 … x_d,

for some 
    ρ'( f̂^𝒞l(ω_1,ω_2,...,ω_d), e^ e_1 ω_1 x_1, …, e^ e_d ω_d x_d) ∈Π( f̂^𝒞l(ω_1,ω_2,...,ω_d), e^ e_1 ω_1 x_1, …, e^ e_d ω_d x_d).

We can expand the exponentials in (<ref>) and (<ref>) to write down the general Clifford-Fourier transform in terms of functions α^k and α̂^k

    f̂^𝒞l = ρ_0 - ∑_i=1^d ρ_1(i) + ∑_i<jρ_2(i,j) - ∑_i<j<kρ_3(i,j,k) + …, 
       f = ρ'_0 + ∑_i=1^d ρ'_1(i) + ∑_i<jρ'_2(i,j) + ∑_i<j<kρ'_3(i,j,k) + …,

for some ρ_k and ρ'_k, where

    ρ_0 = α^0…0, 
       ρ_1(i) ∈Π(e_i, α^0… 1(i) … 0), 
       ρ_2(i,j) ∈Π(e_i, e_j, α^0… 1(i,j) … 0), 
       ρ_3(i,j,k) ∈Π(e_i, e_j, e_k, α^0… 1(i,j,k) … 0), 
       …

and

    ρ'_0 = α̂^0…0, 
       ρ'_1(i) ∈Π(e_i, α̂^0… 1(i) … 0), 
       ρ'_2(i,j) ∈Π(e_i, e_j, α̂^0… 1(i,j) … 0), 
       ρ'_3(i,j,k) ∈Π(e_i, e_j, e_k, α̂^0… 1(i,j,k) … 0), 
       ….





We now check whether Clifford-Fourier “analytic signal” given by the restriction of Clifford based Fourier transform only to positive frequencies works for defining the instantaneous amplitude and phase. As we did before we use bracket notation (<ref>), (<ref>) and check whether Fourier transform written in the form similar to  (<ref>) will work. Positive frequency restricted Clifford “analytic signal” of the real-valued function f: ^d → is given by

    f^𝒞l = ρ'_+0 + ∑_i=1^d ρ'_+1(i) + ∑_i<jρ'_+2(i,j) + ∑_i<j<kρ'_+3(i,j,k) + …,

	
with some ρ'_+i:

    ρ'_+0 = α̂_𝒞l+^0…0, 
       ρ'_+1(i) ∈Π(e_i, α̂_𝒞l+^0… 1(i) … 0), 
       ρ'_+2(i,j) ∈Π(e_i, e_j, α̂_𝒞l+^0… 1(i,j) … 0), 
       ρ'_+3(i,j,k) ∈Π(e_i, e_j, e_k, α̂_𝒞l+^0… 1(i,j,k) … 0), 
       …,

where α̂_𝒞l+^j (x) = ⟨ f^𝒞l( ω), α_j( x, ω) ⟩_+.
	

	
	The Clifford algebra valued “analytic signal” f^𝒞l : ^d →𝒞l_0,d() of the real-valued function f: ^d →, where f^𝒞l is given by (<ref>), in general	does not  have as components the corresponding phase-shifted functions f_j, defined as (<ref>), i.e. if we have
	
    f^𝒞l(x) = f(x) + ∑_i e_i f̃_0 … 1(i) … 0 (x)  + ∑_i<j e_i e_j f̃_0 … 1(i,j) … 0 (x) + ∑_i<j<k e_i e_j e_k f̃_0 … 1(i,j,k) … 0(x) + …

	then there exist j∈{0,1}^d such that |f̃_j|  |f_j| for any d>2.
	Therefore there is no Clifford algebra based Fourier transform of the form (<ref>) and (<ref>) for d>2 that will lead us to the phase-shifts f_j as components of e_j.
	
		To prove this theorem it will suffice to demonstrate that we are not able to choose the proper ordering of multiplicative terms in (<ref>) and (<ref>) such that f̃_j = f_j for any j∈{0,1}^d.
	
		Next we simply show that already for the elements of degree 1, i.e. in front of corresponding e_i, we already must have |f̃_0… 1(i) … 0|  |f_0… 1(i) … 0| for some i. 
		First let us write the Clifford-Fourier transform for elements of degree up to 1
		
    f̂^𝒞l(ω)= α^0 … 0 (ω) - e_1 α^10 … 0(ω) - e_2 α^010 … 0(ω) - e_3 α^0010 … 0(ω) - … - e_d α^0 … 01(ω) + ….

		For the elements of degree up to 1 in (<ref>) we may not care about the order of multiplication because f(x) is real valued. Then we show that there is no ordering of multiplicative terms in the inverse formula (<ref>) to obtain the right signs of phase-shifted components in accordance with (<ref>). For simplicity, but without loss of generality, we consider just the combinations of components corresponing to the first three elements e_1, e_2 and e_3.
		In this case potentially we have the four following candidates for the inverse Fourier transform.
		
			
  * Zero elements are flipped: 
			
    c_0(x) =     α̂^0…0(x) + e_1 ⟨f̂^𝒞l(ω), α_10…0(x, ω) ⟩
       + e_2 ⟨f̂^𝒞l(ω), α_010…0(x, ω) ⟩ + e_3 ⟨f̂^𝒞l(ω), α_0010…0(x, ω) ⟩ + ….

			
  * One element is flipped: 
			
    c_1(x) =     α̂^0…0(x) + ⟨f̂^𝒞l(ω), α_10…0(x, ω) ⟩ e_1  
       + e_2 ⟨f̂^𝒞l(ω), α_010…0(x, ω) ⟩ + e_3 ⟨f̂^𝒞l(ω), α_0010…0(x, ω) ⟩ + ….

			
  * Two elements are flipped:
			
    c_2(x) =     α̂^0…0(x) + ⟨f̂^𝒞l(ω), α_10…0(x, ω) ⟩ e_1 
       + ⟨f̂^𝒞l(ω), α_010…0(x, ω) ⟩ e_2 + e_3 ⟨f̂^𝒞l(ω), α_0010…0(x, ω) ⟩ + ….

			
  * Three elements are flipped:
			
    c_3(x) =     α̂^0…0(x) + ⟨f̂^𝒞l(ω), α_10…0(x, ω) ⟩ e_1 
       + ⟨f̂^𝒞l(ω, α_010…0(x, ω) ⟩ e_2 + ⟨f̂^𝒞l(ω), α_0010…0(x, ω) ⟩ e_3 + ….

		
		
		Therefore for c_i serving as inverse we will obtain the components of degree 2 of f^𝒞l by simple subtitution of each term of f̂^𝒞l(ω) from (<ref>) into (<ref>), (<ref>), (<ref>) and (<ref>). That will give us the following hypercomplex components after we restrict the integration only to positive frequencies. 
		
			
  * For c_0 we have
			
    f̃^0_j:|j|=2 =    -e_2 e_1 ⟨α^100 … 0 , α_010 … 0⟩_+ - e_1 e_2 ⟨α^010 … 0 , α_100 … 0⟩_+ 
       -e_3 e_1 ⟨α^100 … 0 , α_0010 … 0⟩_+ - e_1 e_3 ⟨α^0010 … 0 , α_100 … 0⟩_+ 
       -e_3 e_2 ⟨α^010 … 0 , α_0010 … 0⟩_+ - e_2 e_3 ⟨α^0010 … 0 , α_010 … 0⟩_+ + ….

			
			
  * For c_1 we have
			
    f̃^1_j:|j|=2 =    -e_2 e_1 ⟨α^100 … 0 , α_010 … 0⟩_+ - e_2 e_1 ⟨α^010 … 0 , α_100 … 0⟩_+ 
       -e_3 e_1 ⟨α^100 … 0 , α_0010 … 0⟩_+ - e_3 e_1 ⟨α^0010 … 0 , α_100 … 0⟩_+ 
       -e_3 e_2 ⟨α^010 … 0 , α_0010 … 0⟩_+ - e_2 e_3 ⟨α^0010 … 0 , α_010 … 0⟩_+ + ….

			
			
  * For c_2 we have
			
    f̃^2_j:|j|=2 =    -e_1 e_2 ⟨α^100 … 0 , α_010 … 0⟩_+ - e_2 e_1 ⟨α^010 … 0 , α_100 … 0⟩_+ 
       -e_3 e_1 ⟨α^100 … 0 , α_0010 … 0⟩_+ - e_3 e_1 ⟨α^0010 … 0 , α_100 … 0⟩_+ 
       -e_3 e_2 ⟨α^010 … 0 , α_0010 … 0⟩_+ - e_3 e_2 ⟨α^0010 … 0 , α_010 … 0⟩_+ + ….

			
			
  * For c_3 we have
						
    f̃^3_j:|j|=2 =    -e_1 e_2 ⟨α^100 … 0 , α_010 … 0⟩_+ - e_2 e_1 ⟨α^010 … 0 , α_100 … 0⟩_+ 
       -e_1 e_3 ⟨α^100 … 0 , α_0010 … 0⟩_+ - e_3 e_1 ⟨α^0010 … 0 , α_100 … 0⟩_+ 
       -e_2 e_3 ⟨α^010 … 0 , α_0010 … 0⟩_+ - e_3 e_2 ⟨α^0010 … 0 , α_010 … 0⟩_+ + ….

		
	From the rule in (<ref>) we know that with each flipping of 1 in the upper subscript to 0 at the same position in the lower subscript the sign of bracket changes. If we look at f̃^0_j:|j|=2, we see that terms that are components of e_1 e_2, i.e. ⟨α^100 … 0 , α_010 … 0⟩_+ and ⟨α^010 … 0 , α_100 … 0⟩_+ sum up with opposite signs. The same is true for the components of e_1 e_3 and e_2 e_3 – they come all with opposite signs. Therefore case of c_0-type inverse formula is not in accordance with the rule (<ref>). For the case c_1 we see that the components of e_1 e_2 have the same sign as well as components of e_1 e_3, however components of e_2 e_3 have opposite signs. Thus c_1 is not in accordance with (<ref>). After checking the rule for c_2 and c_3 we see that we will always have different signs for some component. Therefore we are not able to order properly the terms in Clifford algebra based Fourier transform and its inverse Fourier transform to correctly restore all the phase-shifted functions.
	



	From the proof of Theorem <ref> it follows that essentially  only commutative hypercomplex algebra (e_i e_j = e_j e_i) is a good candidate to provide us with the phase-shifts (<ref>) by using positive frequency restriction for Fourier transforms of the type (<ref>) and (<ref>). The word “essentially” above means that algebras with two elliptic hypercomplex units that anticommute are still allowed when d=2.



	There were propositions to use Cayley-Dickson algebras to define the Fourier transform (<ref>),  (<ref>) and corresponding hypercomplex “analytic signal” <cit.>, however as far as elements {e_i} of these algebras are non-commutative we will not reconstruct the phase-shifts correctly.


 
 	
 	We may wonder what happens in case d=2. Is there any Clifford algebra based Fourier transform that provides us with the correct phase-shifted components? It seems that the answer should be - yes. And indeed the symmetric quaternionic based Fourier transform does the job. 
 	Let ℚ be the ring of quaternions with the set of basis elements { 1,i,j,k }. The ring of quaternions coincides with the Clifford algebra 𝒞l_0,2( ) that in turn has the set of basis elements {1, e_1, e_2, e_1 e_2 } where i = e_1,  j = e_2,  k = e_1 e_2. The symmetric quaternionic Fourier transform is defined as
 	
    f̂^q ( ω_1, ω_2 ) = ∫_∫_ e^-e_1 ω_1 x_1 f ( x_1,x_2 ) e^-e_2 ω_2 x_2  x_1  x_2, 
       f ( x_1, x_2 ) = 1/(2π)^2∫_∫_ e^e_1 ω_1 x_1f̂^q ( ω_1,ω_2 ) e^e_2 ω_2 x_2  x_1  x_2,

 	which we may rewrite using the notation from (<ref>) as
 	
    f̂^q(ω_1, ω_2) = α^00 - e_1 α^10 - e_2 α^01 + e_1 α^11 e_2, 
       f(x_1, x_2) = ⟨f̂^q, α_00⟩ + e_1 ⟨f̂^q, α_10⟩ + ⟨f̂^q, α_01⟩ e_2 + e_1 ⟨f̂^q, α_11⟩ e_2.

 	The quaternionic “analytic signal” is obtained by restriction of the quaternion Fourier transform to the positive frequencies as we did before
 	
    f^q_a =    ⟨α^00, α_00⟩_+ + ⟨α^10, α_10⟩_+ + ⟨α^01, α_01⟩_+ + ⟨α^11, α_11⟩_+ 
       - e_1 ⟨α^10, α_00⟩_+ + e_1 ⟨α^00, α_10⟩_+ - e_1 ⟨α^11, α_01⟩_+ + e_1 ⟨α^01, α_11⟩_+ 
        - ⟨α^01, α_00⟩_+ e_2 + ⟨α^00, α_01⟩_+ e_2 - ⟨α^11, α_10⟩_+ e_2 + ⟨α^10, α_11⟩_+ e_2 
        + e_1 ⟨α^11, α_00⟩_+ e_2 + e_1 ⟨α^00, α_11⟩_+ e_2 - e_1 ⟨α^10, α_01⟩_+ e_2 - e_1 ⟨α^01, α_10⟩_+ e_2.

 	As we see, the components of quaternionic analytic signal indeed provide us with the correct phase-shifts given by (<ref>). 
 



§ EXAMPLES AND LIMITATIONS






 §.§ Simple example in 3-D



As a direct illustration of how the construction of envelope works, we provide an example in 3-D. The oscillating function that we analyse is given by the cosine wave modulated by a Gaussian window:


    f(x,y,z)     = e^-10x^2 -20y^2 -20z^2cos( 50 x ) cos( 40 y ) cos( 60 z ) 
        = f_x(x) f_y(y) f_z(z).

This particular toy example allows us to separate the variables when calculating the phase-shifted components f_j from (<ref>). In three dimensions we will have in total | { 0,1 }^3 | = 8 shifted functions.

Calculation of some f_j using (<ref>) reduces to the calculation of direct and inverse cosine and sine transforms. For example for the f_100 shifted component, we will have


    f_100(x,y,z)    = f_y(y) f_z(z)/π∫_0^∞∫_-∞^∞ f_x(x) sin( ω_1 (x-x') )   x' ω_1.

Taking into account that f is even in each variable, after expanding the sine of difference, we see that we have to calculate only forward cosine and inverse sine transforms. This double integral can be calculated in semi-analytic form using erf(x) = 2/√(π)∫_0^x e^-t^2 t by

    2/π∫_0^∞sin( ω x ) ∫_0^∞ e^-αx̃^2cos( ω_x x̃) cos( ωx̃)  x̃ω
        = 1/2 i e^-x(ω_x i + α x)[ erf( ω_x/2 - i α x/√(α)) - e^2 ω_x i xerf( ω_x/2 + i α x/√(α)) ].


To illustrate the resulting envelope of function (<ref>), in Fig. <ref> 
we vizualize a point cloud in a cube [0,1]^3 with removed octant where each point is colour coded according to the value of f (the origin with respect to function (<ref>) is shifted by 0.5 in each direction). This method allows visualization of a function “inside” the cube by observing it over the inner faces of the deleted octant. In Fig. <ref> (a) the original f(x,y,z) is plotted, while in Fig. <ref> (b) we plot the envelope function a(x,y,z) obtained from (<ref>) that was computed on the 8 phase-shifted functions similarly to (<ref>).


































 §.§ Amplitude on a smooth graph



Objective of this section is to illustrate amplitude construction for an oscillating function defined over the vertices of a smooth graph or point cloud. The topology of a graph can be captured by a variety of operators like adjacency operator or Laplacian <cit.>. In this example we start by considering a smooth point cloud patch of curved surface, then we construct the corresponding k nearest neighbors graph on top of the points. Finally we embed the graph in  ^2 and calculate amplitude in the transformed space where the points are flattened. As an embedding of a graph in ^2 we rely on the diffusion maps embedding <cit.>.



The graph in this example is constructed as follows. We take a sampled patch on the sphere consisting of 4096 randomly picked points as illustrated by Fig. <ref> (a). The resulting point cloud is inherently 2-dimensional although it is embedded in 3 dimensions. As an oscillating function we consider a sum of two Gaussians modulated by different frequencies that lie on a curved surface. Then, to obtain an associated graph, we consider the k=20 nearest neighbours graph to mesh the point cloud. Then we use the diffusion maps embedding, which is given by the first two eigenvectors of the normalised Laplacian, to embed the patch in ^2. The resulting embedding of the corresponding k nearest neighbors graph with the same oscillating process is shown in Fig. <ref> (b).

To calculate the discrete version of hypercomplex Fourier transform f̂(ω), we applied the transformation (<ref>) to the discrete oscillating function given in Fig. <ref> (b) (axes are rescaled by a factor of 10). Then corresponding phase-shifted components were constructed by first restricting the spectrum f̂(ω) to positive frequencies according to (<ref>) and  applying inverse Fourier transform (<ref>). All computations were performed in the diffusion space where the initial point cloud is flattened. Discrete version of the Fourier transform (<ref>) was calculated for the first 50 frequencies with step 1. Then we calculate the hypercomplex analytic signal by restricting the spectrum to positive frequencies according (<ref>) prior to applying the inverse Fourier transform. In Fig. <ref> (c) and (d), the envelope function of Eq. (<ref>) is shown over the original point cloud and in the diffusion space, respectively. In Fig. <ref> the phase-shifted components f_j(x) for the four directions j=00, 10, 01 and 11 in the diffusion space are shown. Even though the directions of oscillations are not perfectly aligned to the basis vectors in embedding space (see Fig. <ref> (b)), the resulting amplitude still looks fine. In the following example we clarify this point by looking on how amplitude transforms under rotational transformation of the underlying oscillating process.





 §.§ Amplitude of rotated process



In this section we will look how amplitude transforms under rotational transformation of underlying oscillating process. Let us start by considering the simplest 2-dimensional oscillating process

    f (x, y ) = cos (x) cos (y).

Intuitively it is clear that such process should have constant amplitude. And indeed using definition (<ref>), we see that

    a(x,y) = √(cos (x)^2 cos (y)^2 + sin (x)^2 cos (y)^2 + sin (x)^2 cos (y)^2 + sin (x)^2 sin (y)^2) = 1.

Let us consider the rotation by π/4 applied to f. In this case the coordinates are transformed according to 


    x' = 1/√(2) x - 1/√(2) y,   y' = 1/√(2) x + 1/√(2) y.

By expanding f'(x', y') ≡ f(x', y') we get

    f'(x, y) = cos( 1/√(2) x - 1/√(2) y ) cos( 1/√(2) x + 1/√(2) y ) = 
    1/2cos( √(2) x ) + 1/2cos( √(2) y ).


The phase-shifted components for f' are (recall that Hilbert transform of constant function is zero)

    f'_00 (x, y) =    1/2cos( √(2) x ) + 1/2cos( √(2) y ), 
    
    f'_10 (x, y) =    1/2sin( √(2) x ), 
    
    f'_01 (x, y) =    1/2sin( √(2) y ), 
    
    f'_11 (x, y) =     0.

Combining the terms together we get a'(x,y)

    a'(x,y)^2 =     f'_00 (x, y)^2 + f'_10 (x, y)^2 + f'_01 (x, y)^2 + f'_11 (x, y)^2 
    
    =    1/4cos( √(2) x )^2 + 1/4cos( √(2) y )^2 + 1/2cos( √(2) x ) cos( √(2) y ) 
     
    +    1/4sin( √(2) x )^2  +  1/4sin( √(2) y )^2 
    
    =    1/2[ 1 + cos( √(2) x ) cos( √(2) y ) ],

which is not constant. Therefore definition of amplitude (<ref>) is only valid for oscillations aligned with the chosen orthogonal directions in ^d. 




 §.§ Amplitude of lower dimensional process



Next we provide an example of amplitude computation for 1-dimensional oscillating process in ^2. We start by considering the simplest 1-dimensional oscillations in ^2, given by

    f(x,y) = cos(x).

The phase-shifted components of f are

    f_00 (x, y) =    cos( x ), 
    
    f_10 (x, y) =    sin( x ), 
    
    f_01 (x, y) =     0, 
    
    f_11 (x, y) =     0,

which gives constant instantaneous amplitude a(x,y) = 1 as expected. We can conclude more generally that definition of amplitude (<ref>) works well when oscillations in some directions are degenerate.

Let us consider the rotated version of f and calculate its amplitude. As in the previous section we apply transformation (<ref>) to f(x,y) and write f' as

    f'(x, y) = cos( 1/√(2) x - 1/√(2) y ) = cos( 1/√(2) x ) cos( 1/√(2) y ) + sin( 1/√(2) x ) sin( 1/√(2) y ).

The phase-shifted components of f' are given by

    f'_00 (x, y) =    cos( 1/√(2) x - 1/√(2) y ), 
    
    f'_10 (x, y) =    sin( 1/√(2) x - 1/√(2) y ), 
    
    f'_01 (x, y) =     -sin( 1/√(2) x - 1/√(2) y ), 
    
    f'_11 (x, y) =    cos( 1/√(2) x - 1/√(2) y ).

Summing up the squares of phase-shifted components we get a'(x,y) = √(2), which is not the expected envelope equal to 1. We observe similar problem with amplitude calculation of a rotated process as in the previous example.




§ CONCLUSION AND FUTURE WORK



In this work we extended analytic signal theory to describe oscillating functions over d-dimensional Euclidean space for the case when oscillations are aligned to d orthogonal directions given by orthonormal basis vectors of ^d. There are several directions in which the presented work could be extended. First of all we need a method for amplitude calculation of a general oscillating process where oscillations are observed in various independent directions, not only in the chosen orthogonal. Second, the extension of hypercomplex analytic signal to manifolds with the properly defined hypercomplex holomorphic structure is needed. Third, the development of multidimensional phase-space or space-frequency analysis tools for the presented hypercomplex analytic signal model, like Wigner-Ville and Segal-Bargmann transforms, will allow to understand better the localization properties of multidimensional oscillating processes. Fourth, development of hypercomplex multidimensional filtering theory for the designed hypercomplex analytic signal will bring us a powerful tool for many practical problems. And finally, as a more distant objective there is a hope to find hypercomplex dynamic equations for multidimensional oscillating processes that arise in nature as well as to develop the associated hypercomplex gauge theory that rely on the definition of instantaneous amplitude and phases. From application point of view we are going to investigate how theory presented in this paper could be applied in the domains where multidimensional oscillations are inherent, e.g. image/video analysis, sound and seismic waves, data oscillations in high-dimensional feature space in machine learning as well as complex oscillating phenomena in quantum field theory.




§ HILBERT TRANSFORM FOR UNIT DISK AND UPPER HALF-PLANE


Let us briefly describe the essence of the one dimensional Hilbert transform by following the lines of <cit.>. The usual way to introduce Hilbert transform is by using the Cauchy formula. If f is holomorphic on ⊂ and continuous on disk  up to the boundary ∂, we can get the value of f(z) from the values of f on the boundary ∂ by Cauchy formula

    f(z) = 1/2 π i∫_∂f(ζ)/ζ - z ζ,    z ∈.

We can express the Cauchy kernel, 
    1/2 π i·ζ/ζ - z ,
 by taking ζ = e^iψ and z = r e^i θ, as following

    1/2 π i·ζ/ζ - z   = 1/2 π·- i e^-i ψ· i e^i ψψ/ e^-i ψ( e^i ψ - r e^i θ) =  1/2 π·ψ/(1 - r e^i(θ-ψ))  = 1/2 π· 1 - r e^-i(θ-ψ)/| 1 - r e^i(θ-ψ)|^2ψ
        = ( 1/2 π·(1 - r cos(θ-ψ) )/| 1 - r e^i(θ-ψ)|^2ψ) + i ( 1/2 π· r sin(θ-ψ) /| 1 - r e^i(θ-ψ)|^2ψ).

If we subtract 1/4 πψ from the real part of the Cauchy kernel, we get the Poisson kernel P_r(θ)

    ( 1/2 π i·ζ/ζ - z) - 1/4πψ   = 1/2 π(  1 - r cos(θ-ψ) /| 1 - r e^i(θ-ψ)|^2 - 1/2) ψ
       = 1/2 π( 1/2 - 1/2r^2/1-2r cos( θ - ψ) + r^2) ψ =: 1/2 P_r ( θ - ψ).

Thus the real part is, up to a small correction, the Poisson kernel. The kernel that reproduces harmonic functions is the real part of the kernel that reproduces holomorphic functions.

We describe briefly the Poisson integral formula. If = { z: |z| < 1 } is the open unit disc in  and ∂ is the boundary of  and there is continuous g : ∂→, then the function u: →, given by

    u ( r e^i θ) = 1/2 π∫_-π^π P_r ( θ - τ) g ( e^i τ)  τ,   0 ≤ r < 1,

will be harmonic on  and has radial limit r → 1^- that agrees with g almost everywhere on ∂.

Suppose now that we are given a real-valued function f ∈ L^2 ( ∂). Then we can use the Poisson integral formula to produce a function u on  such that u = f on ∂. Then we can find a harmonic conjugate u^† of u, such that u^†(0) = 0 and u + i u^† is holomorphic on . As a final goal we aim to produce a boundary function f^† for u^† and get the linear operator f ↦ f^†.

If we define a function h on  as

    h(z) = 1/2 π i∫_∂f(ζ)/ζ - z ζ,    z ∈,

then h will be holomorphic in . From (<ref>) we know that the real part of h is up to an additive constant equal to the Poisson integral u of f. Therefore (h) is harmonic on  and (h) is a harmonic conjugate of (h). Thus if h is continuous up to the boundary, then we can take u^† = (h) and f^†( e^iθ) = lim_r → 1^- u^†( r e^i θ).

For the imaginary part of the Cauchy kernel (<ref>) under the limit r → 1^- we get

    1/2 π· r sin(θ-ψ) /| 1 - r e^i(θ-ψ)|^21/2 π·sin(θ-ψ) /| 1 - e^i(θ-ψ)|^2 =     sin (θ - ψ)/4 π( 1 - cos (θ - ψ)) 
    = 2 sin( θ - ψ/2) cos( θ - ψ/2)/8 π(cos( θ - ψ/2))^2
    
    =    1/4 π( θ - ψ/2).


Therefore we obtain the Hilbert transform H: f ↦ f^† on the unit disk  as following

    H [ f ] ( e^iθ) = 1/4 π∫_-π^π f ( e^it) ( θ - t/2)   t.

Taylor series expansion of the kernel in (<ref>) gives us

    ( θ/2) = cos( θ/2)/sin( θ/2) = 1 - (θ/2)^2/2! + …/θ/2( 1 - (θ/2)^2/3! + …)
    = 2/θ + E(θ),

where E ( θ) = O ( |θ| ) is a bounded continuous function. Finally for the Hilbert transform on ∂ we can write


    H [ f ] ( e^iθ) = 1/4 π∫_-π^π f ( e^it) ( θ - t/2)   t = 1/4 π∫_-π^π f ( e^it)  2/θ - t  t + 1/4π∫_-π^π f ( e^it) E ( θ - t )   t.

The first integral above is singular at t = θ and the second is bounded and easy to estimate. Usually we write for the kernel of Hilbert transform

    ( θ/2) ≈2/θ - t,

by simply ignoring the trivial error term. Finally in (<ref>) we have defined the Hilbert transform for the boundary of unit disk. Hilbert transform for a unit disk relates harmonic conjugates of a periodic function f defined over ∂. In the followign we briefly outline how to obtain harmonic conjugate for a function defined over ⊂.

The unit disk  can be conformally mapped to the upper half-plane U = {ζ∈: (ζ) > 0 } by the Möbius map <cit.>

    c: → U, 
       ζ↦ i ·1-ζ/1+ζ.

Since the conformal map of a harmonic function is harmonic, we can carry also Poisson kernel to the upper half-plane <cit.>. The Poisson integral equation for U will be

    u ( x + i y ) = ∫_-∞^∞ P ( x-t,y ) f(t)   t,   y>0,

with the Poisson kernel given by

    P ( x,y ) = 1/π·y/x^2 + y^2.

The harmonic conjugate may be obtained from f by taking convolution with the conjugate kernel

    Q ( x,y ) = 1/π·x/x^2 + y^2.

The Cauchy kernel is related to P and Q by the relation i/π z = P(x,y) + i Q(x,y).
Finally we are able to construct the harmonic conjugate on the boudary ∂ U =

    u^†(x,y) = 1/π∫_-∞^∞ f(t) x-t/(x-t)^2 + y^2  t 1/π∫_-∞^∞ f(t) 1/x-t  t.





§ CAUCHY FORMULA BY KETCHUM AND VLADIMIROV


We briefly describe the setting and the resulting Cauchy formula given in <cit.> and <cit.>.
First let us take a bounded region  G ⊆ with piecewise smooth boundary ∂ G. Second let us consider the mapping of G given by l: w ↦ a + b w with a, b ∈_d onto the plane C = { z: z = a+bw, w ∈_d() }⊂_d(), where _d ( ) gives a complexification of _d, i.e. we replace real coefficients with complex in (<ref>). G is mapped to L and ∂ G is mapped to ∂ L correspondingly. Suppose also that the function f(z) is differentiable  <cit.> in some neighbourhood O of the closure L̅ = L ∪∂ L and b is an invertible element of _d(). Then in the plane C we have the following Cauchy formula

    1/2 π i∫_∂ Lf(ζ)/ζ - z ζ = 
    f(z)    if  z ∈ L, 
    
    0    if  z ∈ C ∖L̅.

As it was outlined in Appendix <ref> the Cauchy formula plays a vital role in the definition of Hilbert transform and one can directly extend the Hilbert transform and define it in the unit open disk (upper half-plane) of C.

	The Cauchy formula (<ref>) holds for real Banach algebra _d in which there exists an element i^2 = -ϵ, where ϵ is the unit element in _d. The plane C consists of the elements
	
    C = {x : x = a + u ϵ + v i,  u,v ∈},  a ∈_d.



The Cauchy formula for several hypercomplex variables is written similarly <cit.>. Now we have the region L = L_1 ×…× L_d, where L_j = { z_j = a_j + b_j w_j} is compact in some open region. By the Hartogs's theorem we have that if the function of several variables f(z_1, …, z_d) is analytic with respect to each variable in G_1 ×…× G_d, then it is analytic in G_1 ×…× G_d and we have the general Cauchy formula

    f(z_1, ..., z_d) = 1/(2 π i)^d∫_∂ L_1 ... ∫_∂ L_df(ζ_1, ..., ζ_d)/(ζ_1 - z_1) · ... · (ζ_d - z_d) ζ_1 …ζ_d

for all z ∈ L.




apalike


