Multiresolution Expansion and Approximation Order of Generalized Tempered Distributions

Abstract

Let K (R) be the generalized tempered distributions of-growth with restricted order N 0 , where the function () grows faster than any linear functions as | |. We show the convergence of multiresolution expansions of K (R) in the test function space K (R) of K (R). In addition, we show that the kernel of an integral operator K (R) K (R) provides approximation order in K (R) in the context of shift-invariant spaces.

Multiresolution Expansions of Distributions: Pointwise Convergence and Quasiasymptotic Behavior

“…Using the assumptions (1) (cf. (3)) and ( 2) on M, one verifies [25,26] that for every l ∈ N and |α|, |β| ≤ r, there exists C l > 0 such that…”

“…In dimension n = 1, Pilipović and Teofanov [16] have shown that if f ∈ C r (R) and all of its derivatives up to order r are of at most polynomial growth, then its multiresolution expansion q j f with respect to an rregular MRA converges to f uniformly over compact intervals. Sohn has considered in [25] the analogous result for functions of growth O(e M (kx) ), but his arguments contain various inaccuracies (compare, e.g., his formulas ( 17) and ( 21) with our (3.7) below). We extend those results here to the multidimensional case and for the generalized multiresolution projections q λ,z with uniformity in the parameter z.…”

“…. We mention the references [18,16,25,29], where related results have been discussed. The difference here is that we give emphasis to uniform convergence over bounded subsets of test functions and other parameters, which will be crucial for our arguments in the subsequent sections.…”

“…For distributions of M-growth, we need to impose stronger regularity conditions on the scaling function. We say that the MRA is (M, r)-regular [26,25] if the scaling function from (iv) can be chosen such that φ fulfills the requirement:…”