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

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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.

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Acknowledgements

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

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  1. H. A. Hassan

    Present address: Department of Mathematics, Faculty of Basic Education, PAAET, Adailiyah, Kuwait

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  1. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

    H. A. Hassan

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Hassan, H.A. A Completeness Theorem for a Hahn–Fourier System and an Associated Classical Sampling Theorem. Results Math 74, 34 (2019). https://doi.org/10.1007/s00025-019-0969-z

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