Abstract
In this paper, we propose an interpolation formula for periodic functions. This formula can be regarded as an analog of the Sinc approximation, which is an interpolation formula for functions defined on the entire infinite interval. Theoretical error analysis and numerical examples show that the proposed formula converges exponentially for analytic periodic functions.
Introduction
The trapezoidal formula with equal mesh is an elementary numerical integration formula, and it works very well for the integral of an analytic function over the entire infinite interval and the integral of an analytic periodic function over an interval of its period length, that is, the formula
Introduction
is efficient if f (x) is an analytic function on the entire infinite interval (−∞, +∞), and
Introduction
is efficient if f (x) is an analytic periodic function of period a. Especially, the formula (1) over the entire infinite interval is widely used in science and engineering combined with the technique of a variable transform [9] . By the way, there exists an interpolation formula with equal mesh over the entire infinite interval. It is the Sinc approximation [7] f
Introduction
sin[(π/h)(x − kh)] (π/h)(x − kh) .
Introduction
It is an interpolation formula with sampling points kh ( k ∈ Z ), that is,
Introduction
it works very well especially if f (x) is an analytic function over the entire infinite interval (−∞, +∞), and we obtain the trapezoidal formula over the entire infinite interval by integrating (3) formally term by term, that is,
Introduction
As the trapezoidal formula for numerical integration over the entire infinite interval (−∞, +∞), the Sinc approximation is applied to various numerical computations combined with the technique of variable transforms such as indefinite integrals [3] , initial value problems of ordinary differential equations [5] , boundary value problems of ordinary differential equations [8] and integral equations [4, 6] . Taking into account that 1. the trapezoidal formula works very well for analytic functions over the entire infinite interval and analytic periodic functions, and 2. the Sinc approximation is an interpolation formula over the entire infinite interval which generates the trapezoidal formula over the entire infinite interval, it is a natural question whether there exists an interpolation formula f (x) ≃ L [f ](x) for periodic functions f (x) with equal mesh which generates the trapezoidal formula for periodic functions, that is,
Introduction
where a is the period of f (x), and
Introduction
This paper gives the answer "yes" to this question, that is, there exists an interpolation formula f (x) ≃ L [f ](x) for periodic functions which satisfies the properties (4) and (5) . The remainder of this paper are structured as follows. In Section 2, we propose an interpolation formula for periodic functions which we desire. In Section 3, we show a theorem on the exponential convergence of the proposed method for periodic analytic functions. In Section 4, we show the effectiveness of the proposed method by some numerical examples. In Section 5, we show concluding remarks.
Sinc approximation for periodic functions
Let f (x) be a periodic function of period a. We propose the following interpolation formula for f (x).
Sinc approximation for periodic functions
(6) We can easily check that
Sinc approximation for periodic functions
, that is,
Sinc approximation for periodic functions
3. The trapezoidal formula for f (x) over (0, a) with an equal mesh h = a/N is obtained by integrating
Sinc approximation for periodic functions
4. The limit as N → ∞ and a → ∞ with h = a/N fixed gives the Sinc approximation
Sinc approximation for periodic functions
The first and second properties are satisfied also by the Sinc approximation. We call the approximation (6) the Sinc approximation for periodic functions.
Sinc approximation for periodic functions
Derivation We show here how to find the Sinc approximation for periodic functions (6) . We begin with the Fourier transform of the Sinc approximation over the entire infinite interval
Sinc approximation for periodic functions
where
Sinc approximation for periodic functions
We compare (7) with the Fourier transform of f (x)
Sinc approximation for periodic functions
We find that
Sinc approximation for periodic functions
is non-zero only in the interval (−π/h, π/h) and, in (7), the integral on the right-hand side of (8) is replaced with its approximation by the trapezoidal formula of equal mesh h. Let f (x) be a periodic function of period a and expand it to the Fourier series
Sinc approximation for periodic functions
where
Sinc approximation for periodic functions
From the above observation of the Sinc approximation over the entire infinite interval, we expect that we can obtain the Sinc approximation of the periodic function f (x) by truncating the infinite sum on the right-hand side of (8) and replace the integral on the right-hand side of (10) by its approximation by the trapezoidal formula. First, we derive the Sinc approximation of the case that the number of sampling points is odd. We truncate the right-hand side of (9) as
Sinc approximation for periodic functions
Then, we approximate the integral on the right-hand side of (10) by its approximation by the (2N 0 + 1)-point trapezoidal formula
Sinc approximation for periodic functions
Substituting (12) into (11), we have
Sinc approximation for periodic functions
Using the formula
Sinc approximation for periodic functions
we obtain the Sinc approximation (6) for N = 2N 0 + 1. Second, we derive the Sinc approximation for the case that the number of sampling points is even. We truncate the Fourier series (9) as
Sinc approximation for periodic functions
Then, we approximate the integral on the right-hand side of (10) by its approximation by the (2N 0 )-point trapezoidal formula
Sinc approximation for periodic functions
Substituting (14) into (13), we have
Sinc approximation for periodic functions
(15) Using the formula
Sinc approximation for periodic functions
we obtain the Sinc approximation (6) for N = 2N 0 .
Convergence theorem
The following theorem shows that the Sinc approximation for periodic functions converges exponentially as the number of sampling points N increases if the function f (x) is an analytic periodic function.
Theorem 1
We suppose that f (z) is a periodic function of period a and analytic in a complex domain containing the strip | Im z| ≦ d. Then we have the inequality
Theorem 1
where
Theorem 1
and C d,a is a positive constant depending d and a only.
Theorem 1
We compare this theorem with the following theorem for the Sinc approximation over the entire infinite interval (Theorem 3.1.3 in [7] ).
Theorem 1
We suppose that a function f (z) is analytic in the strip domain
Theorem 1
Then, we have
Theorem 1
This theorem shows that the error of the Sinc approximation is of order O[exp(−πd/h)] as the mesh h goes to zero. On the other hand, the error of the approximation (6) for periodic analytic functions is of order O[exp(−πd/h)] where h is the mesh size h = a/N . Therefore, the error of the Sinc approximation for periodic functions is of the same order as that of the Sinc approximation over the entire infinite interval.
Theorem 1
Proof of Theorem 1 We prove Theorem 1 only for N = 2N 0 + 1. We can prove the theorem for N = 2N 0 similarly. Since
Theorem 1
we have
Theorem 1
Each term of the first term on the right-hand side of (18) 
Theorem 1
Then, we have
Theorem 1
As for the second term on the right-hand side of (18), since
Theorem 1
Therefore, we have (16) for N = 2N 0 + 1.
Numerical examples
We evaluated the errors of the N -point Sinc approximation and the N -point Chebyshev interpolation
Numerical examples
where
Numerical examples
is the Sinc approximation or the Chebyshev interpolation, of the periodic functions with period a = 2π
Numerical examples
The maximum on the right-hand side of (19) was actually evaluated on 1024 equidistant points in the interval [0, 2π). All the computations were performed using programs coded in C++ and multi-precision arithmetics with 100 decimal digit precision by the software exflib [2] . Figure 1 shows the errors of the two methods applied to the functions (20). From these figures, the Sinc approximation is superior to the Chebyshev interpolation and converges exponentially as the number of sampling points N increases. We evaluated the decay rate of the error ǫ N of the Sinc approximation using the fit command of the software gnuplot. Table 1 shows the results. We can also estimate the decay rate of the error ǫ N by using Theorem 1 for the functions (2) and (3) Table 1 . From this table, the theoretical error estimate make a good agreement with the numerical result especially for the function (2).
Numerical examples
1.0e-120
Numerical examples
1.0e-100
Numerical examples
1.0e-080
Numerical examples
1.0e-060
Numerical examples
1.0e-040
Numerical examples
1.0e-020 
Summary
In this paper, we proposed an approximation of periodic functions, which can be regarded as an analog of the Sinc approximation for functions over the entire infinite interval in the sense that it has equidistant sampling points and it gives the trapezoidal formula with equal mesh by integrating it. From theoretical error As an application of the proposed approximation, it is expected that it can be applied to the boundary element method for boundary value problems of partial differential equations.

{'paper_id': '202537448', 'title': 'An analog of the Sinc approximation for periodic functions', 's2_url': 'https://api.semanticscholar.org/CorpusID:202537448', 'source': 'S2ORC'}