A dyadic analogue of Wiener's Tauberian theorem and some related questions

Abstract

A dyadic analogue is proved of Wiener's Tauberian convolution theorem for two functions. Closedness criteria are established for the linear span of the set of binary shifts for a given function or . A consequence of these criteria is that the linear span of the set of binary shifts for a given function () is dense in the space () if and only if all the Fourier coefficients of with respect to the orthonormalized Walsh system on are non-zero.