Quadratic-phase Fourier transform of tempered distributions and pseudo-differential operators
Manish Kumar, Tusharakanta Pradhan
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.
Topics & Concepts
MathematicsParseval's theoremMathematical analysisFourier transformQuadratic equationFourier integral operatorFourier inversion theoremDistribution (mathematics)Pure mathematicsFourier analysisOperator theoryFractional Fourier transformGeometryMathematical Analysis and Transform MethodsImage and Signal Denoising MethodsDigital Filter Design and Implementation