Behavior of Partial Sums of Wavelet Series

Abstract

Given a distribution f belonging the Sobolev space H 1Â2 , we show that partial sums of its wavelet expansion behave like truncated versions of the inverse Fourier transform of f . Our result is sharp in the sense that such behavior no longer happens in general for H s if s<1Â2. Academic Press

Convergence and Characterization of Wavelets Associated with Dilation Matrix in Variable Spaces

“…The results of Kelly et al [11,12] only assumed that the wavelets are bounded by radial decreasing 1 -functions. The behavior of wavelet expansions outside the Lebesgue set is discussed by Reyes [15], for 1 ≤ < ∞, whereas Kelly et al [11] proved that the wavelet expansion of a function in spaces converges pointwise everywhere on the Lebesgue set of a given function. Tao [16] has extended the results of Meyer [9] and Kelly et al [11,12] and showed that the wavelet expansion of any -function converges pointwise almost everywhere under the wavelet projections, hard sampling, and soft sampling summation methods, for 1 < < ∞.…”