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Results in Mathematics

, 74:34 | Cite as

A Completeness Theorem for a Hahn–Fourier System and an Associated Classical Sampling Theorem

  • H. A. HassanEmail author
Article
  • 17 Downloads

Abstract

We introduce a completeness theorem for a Hahn–Fourier-type trigonometric system, where the sine and cosine functions are replaced by \(q,\omega \)-trigonometric functions, where \(0<q<1,0<\omega \) are fixed. The completeness is established in an appropriate \(L^2\)-space, defined in terms of Jackson–Nörlund integrals. We then derive a \(q,\omega \)-counterpart of the celebrated sampling theorem of Whittaker (Proc R Soc Edinb 35:181–194, 1915), as reported by Kotel’nikov (in: Material for the first all union conference on questions of communications, Moscow, 1933) and Shannon (Proc IRE 37:10–21, 1949), for finite Hahn–Fourier-type integral transforms. A convergence analysis is established and comparative numerical examples are exhibited.

Keywords

Hahn difference operator Jackson–Nörlund transform sampling theory 

Mathematics Subject Classification

39A10 44A55 94A20 

Notes

Acknowledgements

The author thank the anonymous referee for his careful reading of the paper and for the constructive comments.

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt
  2. 2.Department of Mathematics, Faculty of Basic EducationPAAETAdailiyahKuwait

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