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Konvergenzverhalten der konjugierten Shannonschen Abtastreihe (Convergence behavior of the conjugate Shannon sampling series). (German) Zbl 0931.42024

Summary: This paper deals with a question proposed by F. Schipp, regarding the convergence behavior of the conjugate Shannon sampling series for non-bandlimited functions. The convergence behavior is characterized according to the Sobolev spaces \(H^s(\mathbb R)\). If \(s>\frac 12\) then the sequence of the conjugate Shannon sampling series is uniformly convergent for all elements of the space \(H^s(\mathbb R)\). If \(s<\frac 12\) then a function \(f_1 \in H^s (\mathbb R)\) will be constructed, so that the sequence of the conjugate Shannon sampling series is divergent everywhere. The Hilbert transform \(f_1\) is a continuous function.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A05 Interpolation in approximation theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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