We study the approximation of the inverse wavelet transform using Riemannian sums.We show that when the Fourier transforms of wavelet functions satisfy some moderate decay condition,the Riemannian sums converge to the function to be reconstructed as the sampling density tends to infinity.We also study the convergence of the operators introduced by the Riemannian sums.Our result improves some known ones.
The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.
In this paper, we give a characterization of shift-invariant subspaces which are also invariant under additional non-integer translations. Both principal and finitely generated shift-invariant subspaces are studied. Our results improve some known ones.
