Asymptotic Series of General Symbol of Pseudo-Differential Operator Involving Fractional Fourier Transform

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 ofmathematical sciences and engineering. The fractional Fourier transform can be considered as a rotation by an angle α in timefrequency plane and is also called rotational Fourier transform or angular Fourier transform. The fractional Fourier transform [2, 3], with angle α of a function u(x), is defined by


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 timefrequency 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 (R) 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  1 and  2 are constants and  1 () and  2 () are two input functions.
Let   (R) denote the class of measurable functions  defined on R such that where 1 ≤  < ∞.
The convolution of two functions  ∈  1 (R) and  ∈  1 (R) is defined [5,7] as provided that the integral exists.
Let G be a class of all measurable complex-valued functions (, ) which are defined on R × R − {0}.Then, we assume the following properties.
and is bounded to mesaurable function.
Then, we define a function where (  ) ∞ 0 is a sequence of positive real numbers such that   → ∞ as  → ∞.
The global estimate of the above defined function (, ) and of remainders of order  is given as are satisfied for  = 1, 2, 3 . ... In particular the estimates are as follows: where Proof.The proof of above theorem is also obvious by using the same arguments from [6, pages 133-135].

Asymptotic Expansion of Pseudo-Differential Operator Associated with General Symbol
where  ∈ (R),  ∈ R.
Proof.We have By linearity of fractional Fourier transform (5) we get Now using ( 1) and ( 22), we get the required result.
Theorem 9. Let (, ) ∈ G be a symbol and (, ) the associated operator; then one has the following relation: where  ∈ (R),  ∈ R.