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

YILMAZ Y., Erdoğan B., Levent H.

Mathematics, cilt.12, sa.19, 2024 (SCI-Expanded, Scopus) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 12 Sayı: 19
  • Basım Tarihi: 2024
  • Doi Numarası: 10.3390/math12192982
  • Dergi Adı: Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Anahtar Kelimeler: 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
  • Erciyes Üniversitesi Adresli: Evet

Özet

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 (Formula presented.) of (Formula presented.) -valued square-integrable functions, where (Formula presented.) is the class of all compact subsets of complex numbers. Furthermore, we defined the quasi-Paley–Wiener space, (Formula presented.), using the Fourier transform defined for set-valued functions and thus we showed that the sequence (Formula presented.) form also a basis for (Formula presented.). We call this result Shannon’s sampling theorem for set-valued functions. Finally, we gave an application based on this theorem.