A New Class of Pseudo-Differential Operators Involving Linear Canonical Transform
- PDF / 1,157,076 Bytes
- 23 Pages / 439.37 x 666.142 pts Page_size
- 51 Downloads / 187 Views
A New Class of Pseudo-Differential Operators Involving Linear Canonical Transform Manish Kumar1 Accepted: 12 October 2020 © Springer Nature India Private Limited 2020
Abstract The primary aim is to develop a new class of pseudo-differential operators involving Linear Canonical Transform (LCT) and some of its essential results in Schwartz space and tempered distribution space. The secondary aim is to find applications of the LCT while constructing new general partial differential equations (Heat and Wave) and derive their solutions. Further, we have discussed particular cases of these partial differential equations. Furthermore, we have also shown the resolution of the solution of these equations graphically. Keywords Pseudo-differential operators · Linear canonical transform · Fourier transform · Fractional Fourier transform · Schwartz space Mathematics Subject Classification 46F12 · 47G30 · 35K05 · 35L05
Introduction Various linear integral transforms (for instance, fractional Fourier transform (FrFT) [3–6], Fourier transform (FT) [1,2], and Fresnel transform etc.) can be extended to a LCT. It has wide applications in fields such as quantum theory, signal processing, image processing, and to name a few. One can learn some history of LCT from [7]. Due to a quadratic-phase kernel, it is also called as a quadratic-phase transform. It was discovered in paraxial optics by authors [8] and also in quantum mechanics by authors [9,10] simultaneously and independently during the early 1970s. Many authors [11–16] have explored LCT in different names in different contexts. In [11] they referred to as the quadratic phase system. Generalized Huygens integral was defined in [12]. Similarly, in [13] it is addressed as generalized Fresnel transform, and special affine transforms in [14]. For extended FrFT one can see [15] and Moshinskey-Quesne transform in [16] etc. LCT for a function (signal) ϕ ∈ L 1 (R), is defined by CM (x, ξ )ϕ(x)d x, ξ ∈ R, (1) (CM ϕ)(ξ ) = R
B 1
Manish Kumar [email protected] ; [email protected] Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad, Telangana 500078, India 0123456789().: V,-vol
123
165
Page 2 of 23
Int. J. Appl. Comput. Math
where the kernel
and AM = given by:
√
CM (x, ξ ) = AM e 1/B
e−iπ/4 .
ϕ(x) =
,
The corresponding inversion formula exists for non-zero B and is
−1 (CM ϕM (ξ ))(x)
=
where
A∗M
A 2 2 2 iπ D B x − B xξ + B ξ
(2020) 6:165
R
CM (x, ξ )ϕM (ξ )dξ, x ∈ R.
−iπ D x 2 −
2
xξ + A ξ 2
(2)
B B B CM (x, ξ ) = A∗M e , √ AB ; M ∈ S L(2, F) (the special linear group of = −1/B e−iπ/4 , and M =
C D
degree 2 over a real field F) is the set of 2 × 2 matrices with determinant 1. cos α sin α Case 1 If one can take entries of the matrix as M = , where α = nπ and − sin α cos α n ∈ Z; then it reduces to the FrFT as follows: (Csin α,cos α ϕ)(ξ ) = Csin α,cos α (x, ξ )ϕ(x)d x, ξ ∈ R, (3) R
where the kernel
and Asin α,cos α
α 2 2 cos α 2 iπ cos sin α x − sin α xξ + si
Data Loading...
