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arXiv · 2405.17943

Shift-invariant subspaces of Sobolev type

Aleksandar Aksentijević·Suzana Aleksić
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Abstract

This paper has the characteristics of a review paper in which results of shift-invariant subspaces of Sobolev type are summarized without proofs. The structure of shift-invariant spaces $V_s$, $s\in\mathbb{R}$, generated by at most countable family of generators, which are subspaces of Sobolev spaces $H^s(\mathbb{R}^n)$, are announced in \cite{aap} and Bessel sequences, frames and Riesz families of such spaces are characterized. With the Fourier multiplier $\left(1-\frac{\Delta}{4\pi^2}\right)^{s/2}f=\mathcal{F}^{-1}\big((1+|t|^2)^{s/2}\widehat{f}(t)\big)$, we are able to extend notions and theorems in \cite{MB} to spaces of the Sobolev type.

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Aleksandar Aksentijević, Suzana Aleksić. 2024-05-28. Shift-invariant subspaces of Sobolev type. https://arxiv.org/abs/2405.17943

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