Convolution equations in spaces of sequences with an exponential growth constraint

One describes the sets of the solutions of the convolution equations S*x=0 (on the set or on ₊={n:n0}) in the spaces of sequences of the type X=^X(, ), where. One proves that any 1-invariant subspace E,EX, coincides with KezS for some S and, after the Laplace transform can be represented in the form f·A(K_{(, )}), where K_{(, )}={z:<z<}. The space E can be written in the form E=span{{n^kn}_{n z} ^: }+{xX:x_k=0, k, if and only if the representing function t is a pure Weierstrass product (in the ring K_{(, )}, whose zeros do not accumulate to the circumference ¦¦=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSP, Vol. 149, pp. 107–115, 1986.The author expresses his sincere gratitude to N. K. Nikol'skii for the formulation of the problem and for his interest in the paper.