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Recent developments in the theory of the fractional Fourier and linear canonical transforms. (English) Zbl 1143.42004

Fractional calculus is a flourishing field of active research. In this paper the authors restrict themselves to the fractional Fourier operator and friends that are traditionally used in optics, mechanical engineering and signal processing. Because this field is still in full expansion the authors summarized in this survey paper some of the recent developments that appeared in the literature since then, revealing some unexplored aspects.
The paper consists of 14 parts: 1. Introduction; 2. The fractionalization of a linear operator; 3. The fractional Fourier transform.
In this part of the paper the main definitions are given and the main literature of known results on fractional Fourier transforms and his applications are pointed out.
Part 4 is devoted to the theme of linear canonical transforms which are characterized by general matrix A. Under definite elements of the given matrix special cases are set: The Fresnel transform; Dilation operation; Gauss-Weierstrass transform or chirp convolution: Multiplication by a Gaussian or chirp multiplication.
Part 5 is devoted to the geometric interpretation.
Parts 6, 7, 8 are devoted to the fractional operations in which operations like convolution, correlation, \(x\)-shift, \(\xi\)-shift etc. can be defined in the fractional domain.
In Part 9, multidimensional linear canonical transform is defined. In Part 10, fractional transforms in the case of circular symmetry are defined. In general, the canonical transforms for such a situation are known as radial canonical transforms.
In Part 11, fractional angular transform (cyclic transform): the Hilbert transform; the cosine; Sine and Hartley transforms; the Bragman transform; the bilateral Laplace transform; offset transforms.
Part 12 is devoted to the definition of discrete fractional transforms: discrete fractional Fourier transform; discrete cosine; sine and Hartley transform and general LCT (linear canonical transform); discrete Wigner distribution.
In Part 13, applications of the FrFT are used as a modelling tool in quantum mechanics and in optics systems; Filtering; Compression; Image encryption; Neural networks and pattern recognition; edge detection; antennas, radar and sonar; communication theory; tomography.
The references contain 133 titles.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
65T50 Numerical methods for discrete and fast Fourier transforms
26A33 Fractional derivatives and integrals
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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Full Text: Euclid