Affine Systems that Span Lebesgue Spaces

Abstract

We establish rather weak conditions on \(\psi\in L^p(R^d)\) under which the small scale affine system \(\{\psi(a_jx-k): j>0,k\in Z^d\}\) spans \(L^p(R^d), 1\le p<\infty\). The conditions are that the periodization of |ψ| be locally in Lp, that \(\int_{ R^d}\psi dx\not= 0\), and that the dilation matrices aj are expanding, meaning \(\Vert a_j^{-1}\Vert\rightarrow 0 \textrm {as} j\rightarrow\infty\). The periodization of ψ need not be constant; that is, the integer translates \(\{\psi(x-k): k\in Z^d\}\) need not form a partition of unity. The proof involves explicitly approximating an arbitrary function f using a linear combination of the \(\psi(a_jx-k)\), with the coefficients in the linear combination being local average values of f .