### Pseudo-differential operator associated with the fractional Fourier transform

#### 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.

#### Full Text:

PDFISSN: 1331-0623 (Print), 1848-8013 (Online)