", in which case the is usually times . Although these two forms are equivalent, one must be careful about which definition the author uses. For any real , the -angle fractional Fourier transform of a function ƒ is denoted by and defined by Formally, this formula is only valid when the input function is in a sufficiently nice space (such as L1 or Schwartz space), and is defined via a density argument, in a way similar to that of the ordinary Fourier transform (see article), in the general case. If is an integer multiple of π, then the cotangent and cosecant functions above diverge. However, this can be handled by taking the limit, and leads to a Dirac delta function in the integrand. More directly, since must be simply or for an even or odd multiple of respectively. For , this becomes precisely the definition of the continuous Fourier transform, and for it is the definition of the inverse continuous Fourier transform. The FRFT argument is neither a spatial one nor a frequency . We will see why it can be interpreted as linear combination of both coordinates . When we want to distinguish the -angular fractional domain, we will let denote the argument of . Remark: with the angular frequency ω convention instead of the frequency one, the FRFT formula is the Mehler kernel, Properties The -th order fractional Fourier transform operator, , has the properties: Additivity For any real angles , Linearity Integer Orders If is an integer multiple of , then: Moreover, it has following relation Inverse Commutativity Associativity Unitarity Time Reversal Transform of a shifted function Define the shift and the phase shift operators as follows: Then that is, Transform of a scaled function Define the scaling and chirp multiplication operators as follows: Then, Notice that the fractional Fourier transform of cannot be expressed as a scaled version of . Rather, the fractional Fourier transform of turns out to be a scaled and chirp modulated version of where is a different order. Fractional kernel The FRFT is an integral transform where the α-angle kernel is Here again the special cases are consistent with the limit behavior when approaches a multiple of . The FRFT has the same properties as its kernels : symmetry: inverse: additivity: Related transforms There also exist related fractional generalizations of similar transforms such as the discrete Fourier transform. The discrete fractional