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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Yılmaz Yılmaz & Bağdagül Kartal Erdoğan & Halise Levent, 2024. "Shannon’s Sampling Theorem for Set-Valued Functions with an Application," Mathematics, MDPI, vol. 12(19), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:2982-:d:1485519
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