The explicit solutions for a class of fractional Fourier–Laplace convolution equations

Abstract

In this paper, two types of fractional Fourier–Laplace convolutions are defined, and the corresponding fractional Fourier–Laplace convolution theorems associated with the fractional cosine transform (FRCT), fractional sine transform (FRST) and Laplace transform (LP) are investigated in detail. The relationship between the fractional Fourier–Laplace convolutions and the existing convolutions is given, and Young's type theorem as well as the weighted convolution inequality are also obtained. As an application for fractional Fourier–Laplace convolution, the filter design and the system of convolution-type integral equations are also considered, the computational complexity for the multiplicative filter is analysed, and explicit solutions for these equations are obtained.

Keywords:

• Fractional Fourier transform
• Laplace transform
• convolution theorem
• filter
• convolution equation

Mathematics Subject Classifications:

• 44A35
• 45E10
• 47A30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [Nos. 61861044, 11961072, 62001193], the Natural Science Foundation of Shaanxi Province [Nos. 2022JM-400, 2020JM-547], and the Innovation and Entrepreneurship Training Program for College Students of Yan'an University [D2021154].