%0 Journal Article
%T Asymptotic Series of General Symbol of Pseudo-Differential Operator Involving Fractional Fourier Transform
%A S. K. Upadhyay
%A Anuj Kumar
%A Jitendra Kumar Dubey
%J ISRN Mathematical Analysis
%D 2013
%R 10.1155/2013/501382
%X An asymptotic series of general symbol of pseudo-differential operator is obtained by using the theory of fractional Fourier transform. 1. Introduction Namias [1] introduced fractional Fourier transform which is a generalization of Fourier transform. Fractional Fourier transform is the most important tool, which is frequently used in signal processing and other branches of mathematical sciences and engineering. The fractional Fourier transform can be considered as a rotation by an angle in time-frequency plane and is also called rotational Fourier transform or angular Fourier transform. The fractional Fourier transform [2, 3], with angle of a function , is defined by where The corresponding inversion formula is given by where the kernel Zayed [3] and Bhosale and Chaudhary [4] studied fractional Fourier transform of distributions with compact support. Pathak and others [5] defined the pseudo-differential operator involving fractional Fourier transform on Schwartz space and studied many properties. Our main aim in this paper is to generalize the results of Zaidman [6] and to find an asymptotic series of general symbol of pseudo-differential operator involving fractional Fourier transform. Now we are giving some definitions and properties which are useful for our further investigations. Linearity of fractional Fourier transform is given as where and are constants and and are two input functions. Let denote the class of measurable functions defined on such that where . From [5], generalized Sobolev space involving fractional Fourier transform is defined by and . The convolution of two functions and is defined [5, 7] as provided that the integral exists. Let be a class of all measurable complex-valued functions which are defined on . Then, we assume the following properties.(i) exists for all and is bounded to mesaurable function.(ii)We define , then where is complex-valued function defined on , which is measurable in and for all and satisfies the estimate: where £¿£¿ . Let be a strictly decreasing sequence; that is, as and such that for all , Let be an infinite sequence of function defined on . Then, we define a function where is a sequence of positive real numbers such that as . From (12), it is clear that , for , , and . The global estimate of the above defined function and of remainders of order is given as Theorem 1. Let be a sequence of positive real numbers such that the following inequalities: are satisfied for . In particular the estimates are as follows: Proof. The proof of the above theorem is obvious from [6, pages 233-234]. Theorem 2. Let be a
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2013/501382/