Quadratic-phase Fourier transform of tempered distributions and pseudo-differential operators

Abstract

In this work, the quadratic-phase Fourier transform (QPFT) on Schwartz space is defined and its necessary results are derived, including adjoint formula, Parseval identity, and continuity property. Furthermore, the continuity property on tempered distribution space is discussed, and also an example of the QPFT is provided. The present analysis is applied to develop a new class of pseudo-differential operators and prove an important theorem on Schwartz space. The boundary value problems of generalized partial differential equations (i.e. generalized telegraph, wave, and heat equations) are discussed using QPFT, and graphical plots are provided.

Keywords:

• Quadratic-phase Fourier transform
• pseudo-differential operators
• generalized telegraph equations
• generalized wave equations
• generalized heat equations

AMS CLASSIFICATIONS:

• 43A32
• 47G30
• 35Qxx
• 35L05
• 35K05

Acknowledgments

The authors are grateful to BITS Pilani, Hyderabad Campus, for providing research funds (ID No. 2018PHXF0028H).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Birla Institute of Technology and Science, Pilani [2018PHXF0028H].