Regular compactly supported wavelets in Sobolev spaces

On a Family of Filters Arising in Wavelet Construction

“…As before, our point of view is to emphasize the role of the filter pairs (P N (·; λ), Q N−1 (·; λ)) in this problem. Our results extend some aspects of the observations made in [1]. In particular, they only deal with the norm f 2 =…”

“…These results improve upon similar conclusions derived in [2]. In the fifth section, we review and extend the range of applicability of the results in the recent paper [1] on orthonormal wavelets of compact support with any given regularity in Sobolev spaces, highlighting throughout our discussion the role of the filter pair (P N (·; λ), Q N−1 (·; λ)).…”

“…Hence we conclude that lim j→∞ P j f = f and lim j→−∞ P j f = 0 weakly for all f ∈ H n+1 (R). When n = 0 this conclusion was proved in [1] for the weight function 1 + x 2 . Therefore by a simple change of variables it also holds for W(x; q) when n = 0.…”

“…In this section we follow the paper [1] and study wavelet construction in Sobolev spaces. Such a problem has been previously studied by Tchamitchian, Sweden, and Wang.…”

Analysis of a semi-Lagrangian method for the spherically symmetric Vlasov-Einstein system

“…Such reconstruction operator Π h exists and we can refer to [3], for an example in the context of wavelets multiresolution analysis. In fact in [3], it is shown that we can construct a multiresolution analysis in H s where the orthonormal wavelets basis is compactly supported and have the desired smoothness, and such that the orthogonal projection operator can reproduce all polynomials less or equal to any fixed order, i.e. has the property of high-order approximation or accuracy.…”