Pseudo-differential operator associated with the  fractional Fourier transform

Akhilesh Prasad; Praveen Kumar

Vol. 21 No. 1 (2016)

Regular Articles

Pseudo-differential operator associated with the fractional Fourier transform

Published 2016-01-25

Akhilesh Prasad
Praveen Kumar

Akhilesh Prasad
Indian School of Mines, Dhanbad

Praveen Kumar
Indian School of Mines, Dhanbad

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Abstract

The main goal of this paper is to study properties of the fractional Fourier transform on Schwartz type space $\mathscr{S}_{\theta}$. Symbol class $S_{\rho,\sigma}^{m,\theta}$ is introduced. The fractional pseudo-differential operators (f.p.d.o.) associated with the symbol $a(x,\xi)$ is a continuous linear mapping of $\mathscr{S}_{\theta}$ into itself. Kernel and integral representations of f.p.d.o are obtained. Boundedness property of f.p.d.o. is studied. Application of the fractional Fourier transform in solving generalized Fredholm integral equation is also given.

Keywords

Pseudo-differential operator, Fractional Fourier transform, Schwartz space, Sobolev space, Generalized Fredholm integral equation.

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Author Biography

Akhilesh Prasad

Department of Applied Mathematics

Praveen Kumar

Department of Applied Mathematics