Difference equations in weighted spaces of sequences

Abstract

Let $\varphi= \{\varphi_m\}_{m=1}^{\infty}$ be a family of convex functions $\varphi_m$ on ${\mathbb R}^n$ with certain growth conditions. With a help of restrictions of functions $\varphi_m$ on ${\mathbb Z}^n$ a weighted space of functions on ${\mathbb Z}^n$ denoted as $A_{\varphi}$ is defined. Linear continuous functionals on this space in terms of their Fourier-Laplace transform are described. This description and functional analysis methods allowed to study surjectivity of difference operators on $A_{\varphi}$ and spectral synthesis problem in the kernel of such operators for a special case of a family $\varphi$.