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Session 18. Harmonic analysis, orthogonal expansions and Dunkl theory
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Some results on a subspace of \(L^2(\mathbb R)\) spanned by shifts of a simple function |
Maciej Paluszyński, University of Wrocław, Poland
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Given a closed subspace \(V\subset L^2(\mathbb R)\) spanned by the integer shifts of a single function \(\varphi\) one considers the function \(P_\varphi(\xi)=\sum_{k\in\mathbb Z}|\hat\varphi(\xi+2\pi k)|^2\). It is known that the size properties of this function correspond to properties of the integer shifts of \(\varphi\) as the spanning set. We will discuss some of these correspondences. We will also present the recent solution (by Saliani), which uses a powerful theorem of Kislyakov, of a long standing conjecture (attributed to Weiss): \(P_\varphi>0\) a.e. if and only if the set \(\{\varphi(\cdot-k):k\in\mathbb Z\}\) is \(L^2\)-linearly independent.
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