Multiresolution Expansions of Distributions: Pointwise Convergence and Quasiasymptotic Behavior

Abstract

Abstract. In several variables, we prove the pointwise convergence of multiresolution expansions to the distributional point values of tempered distributions and distributions of superexponential growth. The article extends and improves earlier results by G. G. Walter and B. K. Sohn and D. H. Pahk that were shown in one variable. We also provide characterizations of the quasiasymptotic behavior of distributions at finite points and discuss connections with α-density points of measures.

Multiresolution expansions and wavelets in Gelfand–Shilov spaces

“…It is well known that the effectiveness of an MRA to approximate functions depends on how regular a scaling function could be chosen inside V 0 . In this regard Meyer [12] introduced the notion of r-regular MRA and studied approximation properties in subspaces of S ′ (R d ), see [10,20] for generalizations. The ensuing finer regularity notion is well-suited for our analysis.…”

“…For (i), note that q m ϕ → ϕ pointwise (actually, the pointwise convergence holds under rather mild hypotheses even for a large class of distributions, cf. [10]). Thus, using Lemma 4.2, we only have to check that {q m ϕ} m∈N is a bounded sequence in S s t (R d ).…”

THE POINT WISE BEHAVIOR OF 2-DIMENSIONAL WAVELET EXPANSIONS IN $L^p(R^2)$

“…By using the biorthogonal B-spline wavelets, Junjian in [10] investigate the convergence property of wavelet expansion which is divergencefree and non divergence-free wavelet expansion by using the characterization of vector-valued Besov spaces function. By using the wavelet expansions, the pointwise behavior of Schwartz distributions in several variables is studied in [7], also the characterizations of the quasi asymptotic behavior of distributions at finite points are provided and the connections with α−density points of measures are discussed. In [1] Singh study the point wise convergence of wavelet expansions of L 2 (R) functions, by using prolate spheroidal wavelet.…”

Pointwise Fourier inversion of distributions on spheres

“…A complete characterization for Fourier series and Fourier integrals on R was given in [8]. Note that the pointwise convergence or summability of expansions of distributions has been investigated with respect to other orthogonal systems, such as wavelets (see [5], [9], [10]).…”