Two New Convolutions for the Fractional Fourier Transform

Abstract

In this paper we introduce two novel convolutions for the fractional Fourier transforms (FRFT), and prove natural algebraic properties of the corresponding multiplications such as commutativity, associativity and distributivity, which may be useful in signal processing and other types of applications. We analyze a consequent comparison with other known convolutions, and establish a necessary and sufficient conditions for the solvability of associated convolution equations of both the first and second kind in L 1 (R) and L 2 (R) spaces. An example satisfying the sufficient and necessary condition for the solvability of the equations is given at the end of the paper.

New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

“…In what follows, the notation (F )(x) of the Fourier series will be replaced by  (x) given by (1). Furthermore, the representation in the form (5) naturally suggests that we define a finite Fourier-type transformation as follows.…”

“…Introducing new convolutions has a direct impact in the areas of integral equations and operator theory, and different convolutions are as much important as many properties and applications they will be able to exhibit. [1][2][3][4][5] Although the study of convolutions can also be seen as a classical subarea of mathematical analysis, the reason why it continues to attract the interest of several researchers is due to its high potential in exhibiting new properties and concrete applications. In fact, such convolutions can be seen as integral transforms that we can study from a theoretical point of view.…”

New sampling theorem and multiplicative filtering in the FRFT domain

“…In [3], we have introduced two new convolutions for the FRFT. Namely, one of such convolutions was defined by…”

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

“…For other convolutions and integral operators, while not being exhaustive, we refer the reader to [1,2,3,8,12,13,14,15,16,17,18,22,25,26,28]. In addition, it is relevant to have in mind that the factorization property of convolutions is crucial in solving corresponding convolution type equations [6,7,11,25].…”