Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value belongs to a certain space of smooth elements of the operator A is equivalent to the convergence of a certain weighted sum of integral residuals. As a corollary, we obtain direct and inverse theorems of the theory of approximation in the mean. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 557–563, April, 2008.