Sequences of polynomial functions which converge to solutions of partial differential equations with quasianalytical coefficients are constructed. Estimates of the degree of convergence are also given.
1998 Academic Press 1. INTRODUCTION 0.1 G. Freud and P. Nevai began to develop a theory of weighted approximation for weights on R (see [1,2,8,9]).
In [3], Babin constructed and investigated solutions of differential equations with analytical coefficients using methods of approximation theory. This leads to an idea of extending his results to wider types of infinitely differentiable functions. By a result due to Babin [4], a polynomial approximation u(x)=lim n Γ P n (A) f of the solution u(x) of the differential equation Au(x)= f (x), where A is a semi-bounded, self-adjoint operator, exists if and only if the coefficients of the operator are quasianalytical.
We will consider in this paper differential operators with coefficients which belong to Carleman classes of quasianalytical functions. We recall fundamental definitions and notation (see also [3 5]).
Consider the set of sequences of positive numbers with the rate of growth greater than or equal to (ck) k for some c>0. The sequences [a k ] and [b k ] are said to be equivalent if there are c 1 , c 2 >0 such that, for some number k 0 , a k (c 1 ) k b k , b k (c 2 ) k a k , for k>k 0 .
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