A Completeness Theorem for a Hahn–Fourier System and an Associated Classical Sampling Theorem
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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.
KeywordsHahn difference operator Jackson–Nörlund transform sampling theory
Mathematics Subject Classification39A10 44A55 94A20
The author thank the anonymous referee for his careful reading of the paper and for the constructive comments.
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