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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1059–1092 | Cite as

Fourier-Like Multipliers and Applications for Integral Operators

  • Saifallah GhobberEmail author
Article

Abstract

Timelimited functions and bandlimited functions play a fundamental role in signal and image processing. But by the uncertainty principles, a signal cannot be simultaneously time and bandlimited. A natural assumption is thus that a signal is almost time and almost bandlimited. The aim of this paper is to prove that the set of almost time and almost bandlimited signals is not excluded from the uncertainty principles. The transforms under consideration are integral operators with bounded kernels for which there is a Parseval Theorem. Then we define the wavelet multipliers for this class of operators, and study their boundedness and Schatten class properties. We show that the wavelet multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator. Moreover we prove that a signal which is almost time and almost bandlimited can be approximated by its projection on the span of the first eigenfunctions of the phase space restriction operator, corresponding to the largest eigenvalues which are close to one.

Keywords

Multiplier Localization operator Uncertainty principle Nash inequality Carlson inequality 

Mathematics Subject Classification

81S30 94A12 45P05 42C25 42C40 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, College of ScienceKing Faisal UniversityAl-HasaKingdom of Saudi Arabia
  2. 2.LR11ES11 Analyse Mathématiques et Applications, Faculté des Sciences de TunisUniversité de Tunis El ManarTunisTunisia

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