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Localization operators and inversion formulas for the Dunkl–Weinstein–Stockwell transform

  • Fethi Soltani EMAIL logo and Ibrahim Maktouf

Abstract

We define and study the Stockwell transform S g associated to the Dunkl–Weinstein operator Δ k , β and prove a Plancherel theorem and an inversion formula. Next, we define a reconstruction function f Δ and prove Calderón’s reproducing inversion formula for the Dunkl–Weinstein–Stockwell transform S g . Moreover, we define the localization operators L g ( σ ) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we introduce and study the extremal function F η , k := ( η I + S g S g ) 1 S g ( k ) , and we deduce best approximate inversion formulas for the Dunkl–Weinstein–Stockwell transform S g on the Sobolev space H k , β s ( R + d + 1 ) .

MSC 2010: 2B10; 44A05; 44A20

Acknowledgements

We thank the referee for the careful reading and editing of the paper.

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Received: 2023-03-06
Revised: 2023-04-08
Accepted: 2023-04-27
Published Online: 2023-11-08
Published in Print: 2024-04-01

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