Characterizing Convergence Rates for Multiresolution Approximations
Mark Kon, Louise Arakelian Raphael · Wavelet analysis and its applications · 1998
Characterizations of convergence rates of multiresolution and wavelet approximations with respect to the supremum norm are given for functions in the L2-Sobolev spaces Hs. Under certain assumptions on the homogeneous Sobolev space Hhs to which the basic wavelet ψ or scaling function ϕ belongs, it is shown that the error at level n for f ∈ Hs is given bysupxfx−Pnfx≤C2−nsfHs, where Pn is the projection onto the scaled space at resolution level n. In other cases, rates of convergence depend on the homogeneous Sobolev class of ϕ or ψ and not on f. Necessary and sufficient conditions on the wavelet, scaling function, and operator I − Pn are provided for a given rate of convergence. Finally, these conditions are related to the Strang-Fix conditions.