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

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–Norlund 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.