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We study the shift-preserving operator \(L:V_s\to V_s\) and the range operator \(R_s\) and their relationship, where \(V_s\) is a shift-invariant subspace of Sobolev space \(H^s(\mathbb {R}^n)\) , \(s\in \mathbb {R}\) . Using the range operator, we give a result about dual frames. For the shift-invariant space \(V_s\) generated by d functions, we find conditions on L and a finite set \(\{\phi _i: \phi _i\in V_s, i=1,\ldots ,m\}\) so that the collection \(\{L^j\phi _i:i=1,\ldots ,m, j =0,\ldots ,d-1\}\) is a frame generator for \(V_s\) .

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Shift-Invariant Subspaces of Sobolev Spaces and Shift-Preserving Operators

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  • 发表日期：

    2025年

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  • Aleksandar Aksentijević,
  • Suzana Aleksić,
  • Stevan Pilipović

摘要

We study the shift-preserving operator \(L:V_s\to V_s\) and the range operator \(R_s\) and their relationship, where \(V_s\) is a shift-invariant subspace of Sobolev space \(H^s(\mathbb {R}^n)\) , \(s\in \mathbb {R}\) . Using the range operator, we give a result about dual frames. For the shift-invariant space \(V_s\) generated by d functions, we find conditions on L and a finite set \(\{\phi _i: \phi _i\in V_s, i=1,\ldots ,m\}\) so that the collection \(\{L^j\phi _i:i=1,\ldots ,m, j =0,\ldots ,d-1\}\) is a frame generator for \(V_s\) .