The structure and the properties of the eigenfunctions of the canonical integral transform are investigated. It is shown that a signal can be decomposed into a set of the orthogonal eigenfunctions of the generalized Fresnel transform. The property that the set contains a finite number of functions is obtained. The canonical integral transform, also known as the generalized Fresnel transform (GF...

Abstract Motivated by the recent studies and developments of integral transforms with various special matrix functions, including orthogonal polynomials as kernels, in this article we derive formulas for Fourier cosine sine functions involving generalized Bessel polynomials. With help these several results are obtained, which extensions corresponding standard cases. The given here general chara...

Hadamard transform is an important tool in discrete signal processing. In this paper, we define the discrete fractional Hadamard transform which is a generalized one. The development of discrete fractional Hadamard is based upon the same spirit of discrete fractional Fourier transform.

The sampling theory is basic and crucial in engineering sciences. On the other hand, the linear canonical transform (LCT) is also of great power in optics, filter design, radar system analysis and pattern recognition, etc. The Fourier transform (FT), the fractional Fourier transform (FRFT), Fresnel transform (FRT) and scaling operations are considered as special cases of the LCT. In this paper,...

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty 1 principle. They both express Fourier dualities within the space of tempered distributions and 2 these dualities are furthermore the inverses of one another. While Poisson’s summation 3 formula expresses a duality between discretization and periodization, Heisenberg’s 4 uncertainty principle expresses a duality be...