Shannon’s Sampling Theorem for Set-Valued Functions with an Application

Abstract

In this study, we defined a kind of Fourier expansion of set-valued square-integrable functions. In fact, we have seen that the classical Fourier basis also constitutes a basis for the Hilbert quasilinear space L2(−π,π,Ω(C)) of Ω(C)-valued square-integrable functions, where Ω(C) is the class of all compact subsets of complex numbers. Furthermore, we defined the quasi-Paley–Wiener space, QPW, using the Fourier transform defined for set-valued functions and thus we showed that the sequence sinc.−kk∈Z form also a basis for QPW. We call this result Shannon’s sampling theorem for set-valued functions. Finally, we gave an application based on this theorem.

Keywords

• inner-product quasilinear spaces
• non-deterministic signals
• Fourier expansion of set-valued square-integrable functions
• Shannon’s sampling theorem for set-valued functions
• Hilbert quasilinear spaces