Uncertainty principles for the quadratic‐phase Fourier transforms. (14th April 2021)

• Record Type: Journal Article
• Authors: Shah, Firdous A.; Nisar, Kottakkaran S.; Lone, Waseem Z.; Tantary, Azhar Y.
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 13(2021)
• Page Start: 10416
• Page End: 10431
• Publication Date: 2021-04-14
• Subjects: quadratic‐phase Fourier transform -- uncertainty principle -- Heisenberg's inequality -- logarithmic and local inequalities -- concentration‐based uncertainty principle -- Hardy's inequality; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.7417
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : The quadratic‐phase Fourier transform (QPFT) is a recent addition to the class of Fourier transforms and embodies a variety of signal processing tools including the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this article, we formulate several classes of uncertainty principles for the QPFT. Firstly, we formulate the Heisenberg's uncertainty principle governing the simultaneous localization of a signal and the corresponding QPFT. Secondly, we obtain some logarithmic and local uncertainty inequalities such as Beckner and Sobolev inequalities for the QPFT. Thirdly, we study several concentration‐based uncertainty principles, including Nazarov's, Amrein–Berthier–Benedicks's, and Donoho–Stark's uncertainty principles. Finally, we conclude the study with the formulation of Hardy's and Beurling's uncertainty principles for the QPFT.