Quantifying invariance properties of shift-invariant spaces (Q2450947)

The distance of an arbitrary translate \(T_r(V(\varphi))\) of a principal shift invariant space \newline \( V(\varphi)=\Bigl\{\sum_{k\in\mathbb{Z}^n} c_k \varphi(\cdot-k),\, \{c_k\}\in\ell^2(\mathbb{Z}^n)\Bigr\}\) to \(V(\varphi)\) is quantified by the shift-variance measure \(\displaystyle TV(S)=\sup_{r\in\mathbb{R}^n}\|[T_r,S]\|_\infty\) where \(T_r f(x)=f(x-r)\), \(S\) is a general bounded linear operator, and \([T,S]\) is the commutator \(TS-ST\). If \(P=P_V\) denotes the orthogonal projection onto \(V=V(\varphi)\) then \(\displaystyle\text{dist}(T_rf,V)=\|T_rf-PT_rf\|_2=\|[T_r,P]f\|_2\leq TV(P)\). It is shown that \(TV(P_\varphi)\geq\sqrt{1-C}\) if it can be shown that \(\|\widehat{\varphi}_\xi\|_4^2\leq C\) for all \(\xi\) in a subset of \([0,1)^n\) of positive measure, where \(f_\xi=\{f(\xi+k)\}_{k\in\mathbb{Z}^n}\). The main theorem then states that if \(\varphi\in L^2(\mathbb{R}^n)\) has orthogonal shifts, is continuous and vanishes at infinity then one can take \(C=1/\sqrt{2}\). A sharper lower bound is established when in addition, the Zak transform of \(\varphi\ast \widetilde\varphi\) (\(\widetilde\varphi(t)=\overline\varphi(-t)\)) is continuous. In that case, \(TV(P_\varphi)=1\), which can be viewed as a sort of generalization of the Balian-Low theorem. This bound applies, in particular, when \(\sum_k\sup |\varphi\chi_{k+[0,1)^n}|<\infty\).