Some Results on the Convergence of Sampling Series Based on Convolution Integrals
Sônia Maria Alves Gomes, Elsa Cortina · SIAM Journal on Mathematical Analysis · 1995
A generalization of sampling series is introduced by considering expansions in terms of scaled translates of a basic function with coefficients given by sampled values of the convolution of a function f with a kernel of Fejér’s type. Such expressions have been used in finite element approximations, sampling theory and, more recently, in wavelet analysis. This article is concerned with the convergence of these series for functions f that exhibit some kind of local singular behavior in time or frequency domains. Pointwise convergence at discontinuity points and Gibbs phenomena are analysed. The convergence in the $H^s $-norm is also investigated. Special attention is focused on multiresolution analysis approximations and examples using the Daubechies scaling functions are presented.