Band-limited scaling and wavelet expansions

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 QjfQjf to f   as j→+∞j→+∞ in the L2L2-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 LpLp, p>1p>1, we assume that there exists δ∈(0,1/2)δ∈(0,1/2) such that φˆ¯φ˜ˆ=1 a.e. on [−δ,δ][−δ,δ], φˆ=0 a.e. on [l−δ,l+δ][l−δ,l+δ] for all l∈Z∖{0}l∈Z∖{0}. For appropriate band-limited or compactly supported functions φ˜, the estimate ‖f−Qjf‖p⩽Cωr(f,2−j)Lp‖f−Qjf‖p⩽Cωr(f,2−j)Lp, where ωrωr denotes the r  -th modulus of continuity, is obtained for arbitrary r∈Nr∈N. For tempered distributions φ˜, we proved that QjfQjf tends to f   in the LpLp-norm, p⩾2p⩾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)φjnQjf=∑n∈ZLf(2−j⋅)(n)φjn. The corresponding wavelet frame-type expansions are found.