Wavelet Bases for Microlocal Filtering and the Sampling Theorem inL_p(Rⁿ)∗

Abstract

An orthonormal wavelet basis in L 2 ðR n Þ used for microlocal filters, which decompose signals into microlocal contents, is shown to be a ''stepwise'' unconditional basis in L p ðR n Þ (1 < p < 1). Other related spaces are also treated. As part of the proof, an elementary proof of the L p version of the sampling theorem with unconditional convergence is given. Finally, an application is given to the expression of some distributions as sums of boundary values of holomorphic functions.

On the unconditional convergence of wavelet expansions for continuous functions

“…∑ ∞ n=1 sin nt n log(1+n) ∈ W 1,1 (Ω) with Ω = (−π, π) its Fourier series does not converge absolutely.…”

“…is characteristic functions of a finite sum of bounded closed intervals (unimodular wavelets)}, {ψ j,k (t)} is an unconditional basis in X =X = L p (R) with 1 < p < ∞ (see [1], [5]).…”

“…Let Ω be an open set of R. The space of bounded variations functions is denoted by BV (Ω) with the norm ∥f ∥ BV := ∥f ∥ L 1 + V (f, Ω), where V is the total variation. The Sobolev space W 1,1 (Ω) is a subspace of BV (Ω) and has the decay property at infinity for the unbounded domain Ω (see e.g., [10]). We remark that BV (Ω) allows jump-type discontinuities.…”

“…We remark that BV (Ω) allows jump-type discontinuities. When Ω is a bounded domain, the similar function spaces BV (Ω) and W 1,1 (Ω) include the space of Lipschitz continuous functions Lip(Ω), but not the space of Hölder continuous functions C α (Ω) with 0 < α < 1. In case of Ω = R, after modifying the function on a set of measure zero we find that Lip(R) = W 1,∞ loc (R), and by the Sobolev embedding theorem the following set inclusions hold:…”

“…(ii) F If f ∈ W 1,1 (Ω)∩C α (Ω) for α > 0, the Fourier series converges uniformly and absolutely, i.e., ∑ j∈Z |c j | < ∞. In fact, W 1,1 (Ω) can be relaxed to BV (Ω).…”