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

Yılmaz Yılmaz, Bağdagül Kartal Erdoğan () and Halise Levent
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Yılmaz Yılmaz: Department of Mathematics, Inonu University, Malatya 44280, Türkiye
Bağdagül Kartal Erdoğan: Department of Mathematics, Erciyes University, Kayseri 38039, Türkiye
Halise Levent: Department of Mathematics, Inonu University, Malatya 44280, Türkiye

Mathematics, 2024, vol. 12, issue 19, 1-14

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 L 2 ( − 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, Q P W , using the Fourier transform defined for set-valued functions and thus we showed that the sequence s i n c . − k k ∈ Z form also a basis for Q P W . 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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