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

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mfenced separators="" open="[" close="]"><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi></mfenced><mo>,</mo><mo>Ω</mo><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula>-valued square-integrable functions, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the class of all compact subsets of complex numbers. Furthermore, we defined the quasi-Paley–Wiener space, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mi>P</mi><mi>W</mi></mrow></semantics></math></inline-formula>, using the Fourier transform defined for set-valued functions and thus we showed that the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="{" close="}"><mi>s</mi><mi>i</mi><mi>n</mi><mi>c</mi><mfenced separators="" open="(" close=")"><mo>.</mo><mo>−</mo><mi>k</mi></mfenced></mfenced><mrow><mi>k</mi><mo>∈</mo><mi mathvariant="double-struck">Z</mi></mrow></msub></semantics></math></inline-formula> form also a basis for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mi>P</mi><mi>W</mi></mrow></semantics></math></inline-formula>. We call this result Shannon’s sampling theorem for set-valued functions. Finally, we gave an application based on this theorem.