Sampling and approximation in shift invariant subspaces of \(L_2(\mathbb{R})\). (English) Zbl 1479.94086
Rassias, Michael Th. (ed.), Harmonic analysis and applications. Cham: Springer. Springer Optim. Appl. 168, 1-19 (2021).
Summary: Let \(\varphi\) be a continuous function in \(L_2(\mathbb{R})\) with a certain decay at infinity and a non-vanishing property in a neighborhood of the origin for the periodization of its Fourier transform \(\widehat{\phi } \). Under the above assumptions on \(\varphi \), we derive uniform and non-uniform sampling expansions in shift invariant spaces \(V_{\phi } \subset L_2(\mathbb{R})\). We also produce local (finite) sampling formulas, approximating elements of \(V_\varphi\) in bounded intervals of \(\mathbb{R} \), and we provide estimates for the corresponding approximation error, namely, the truncation error. Our main tools to obtain these results are the finite section method and the Wiener’s lemma for operator algebras.
For the entire collection see [Zbl 1470.42003].
For the entire collection see [Zbl 1470.42003].
MSC:
| 94A20 | Sampling theory in information and communication theory |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
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