Construction and investigation of solutions of differential equations by methods in the theory of approximation of functions

The steady-state equation $Au_0=f$, the parabolic Cauchy problem $u_1'(t)=Au_1(t)$, $u_1(0)=f$, and the hyperbolic problem $u_2''(t)=Au_2(t)$, $u_2(0)=f$, $u_2'(0)=0$, are considered, where $A$ is a matrix-valued positive selfadjoint second-order partial differential operator with analytic coefficients, and $f$ is an analytic function.
Methods in the theory of weighted approximation of functions by polynomials on the line are used to construct polynomial representations of solutions of these problems of the form $u_i=\lim_{h\to\infty}P_n^i(A)f$, where the polynomials $P_n^i(\lambda)$, $i=0,1,2$, are constructed in explicit form. Estimates of the rate of convergence are given. With the help of these estimates and Bernstein's inverse theorems in approximation theory, theorems are obtained on the smoothness and analyticity of solutions of degenerate systems whose coefficients are trigonometric polynomials.
Bibliography: 9 titles.