Convolution equations in spaces of sequences with an exponential growth constraint

Abstract

One describes the sets of the solutions of the convolution equations S*x=0 (on the set ℤ or on ℤ₊={n∈ℤ:n⩾0}) in the spaces of sequences of the type X=^X(β, ∝), where

. One proves that any 1-invariant subspace E,E⊂X, coincides with KezS for some S and, after the Laplace transform

can be represented in the form f·A(K_{(∝, β)}), where K_{(∝, β)}={z:∝<∣z∣<β}. The space E can be written in the form E=span{{n^kλⁿ}_n∈ ^:_z }+{x∈X:x_k=0, k<m}, δ⊂ℂ, if and only if the representing function t is a pure Weierstrass product (in the ring K_{(∝, β)}, whose zeros do not accumulate to the circumference ¦λ¦=α.