Band-limited scaling and wavelet expansions

Abstract

Operators Qjf=∑n∈Z〈f,φ˜jn〉φjn are studied for a class of band-limited functions φ and a wide class of tempered distributions φ˜. Convergence of Qjf to f as j→+∞ in the L2-norm is proved under a very mild assumption on φ, φ˜, and the rate of convergence is equal to the order of Strang–Fix condition for φ. To study convergence in Lp, p>1, we assume that there exists δ∈(0,1/2) such that φˆ¯φ˜ˆ=1 a.e. on [−δ,δ], φˆ=0 a.e. on [l−δ,l+δ] for all l∈Z∖{0}. For appropriate band-limited or compactly supported functions φ˜, the estimate ‖f−Qjf‖p⩽Cωr(f,2−j)Lp, where ωr denotes the r-th modulus of continuity, is obtained for arbitrary r∈N. For tempered distributions φ˜, we proved that Qjf tends to f in the Lp-norm, p⩾2, with an arbitrary large approximation order. In particular, for some class of differential operators L, we consider φ˜ such that Qjf=∑n∈ZLf(2−j⋅)(n)φjn. The corresponding wavelet frame-type expansions are found.