Publ. Math. Debrecen 40 / 1–2 (1992), 17–33 Solving convolution equations in S 0 + by numerical method By S. PILIPOVI ´ C (Novi Sad) and M. STOJANOVI ´ C (Novi Sad) Abstract. By using expansions of elements from S 0 + into Laguerre series we in- vestigate the convolution equations in this space. We give examples of series expansions and present a numerical method for solving convolution equations. Also, we consider the convolution equations in LG 0 e . 0. Introduction Convolution equations in S 0 + include as special cases a lot of types of differential and integrodifferential equations. This space is a convolution algebra and a natural frame for the extension and the use of the Laplace transformation. In the first part of the paper we give the structural properties of the basic spaces and their duals S 0 + and LG 0 e from the point of view of Laguerre expansions of their elements. Note that the coefficients of f ∈ S 0 + ≡ LG 0 , respectively of f ∈ LG 0 e , expanded into Laguerre series f = ∑ a n l n satisfy ∑ |a n | 2 n -2k < ∞ for some k> 0, respectively ∑ |a n | 2 k -2n < ∞ for some k> 0. By using expansions of elements from S 0 + (LG 0 e ) into Laguerre series we investigate the convolution in it and, in the second part of the paper, the convolution equation f * g = h, where f ∈ S 0 + (LG 0 e ) and h ∈ S 0 + (LG 0 e ) are known. We give examples of series expansions and a numerical method of finding coefficients in the expansion of g. Mathematics Subject Classification : Primary 46F10, 44A35, Secondary 65Jxx, 42C15, 45E10. Key words and phrases : A 0 –type spaces, Laguerre expansion, convolution equations, Laplace transform.