ON APPROXIMATION OF SOLUTIONS OF OPERATOR-DIFFERENTIAL EQUATIONS WITH THEIR ENTIRE SOLUTIONS OF EXPONENTIAL TYPE

We consider an equation of the form y'(t) + Ay(t) = 0, t is an element of [0,8), where A is a nonnegative self-adjoint operator in a Hilbert space. We give direct and inverse theorems on approximation of solutions of this equation with its entire solutions of exponential type. This establishes a one-to-one correspondence between the order of convergence to 0 of the best approximation of a solution and its smoothness degree. The results are illustrated with an example, where the operator A is generated by a second order elliptic differential expression in the space L-2(Omega) (the domain Omega subset of R-n is bounded with smooth boundary) and a certain boundary condition.