Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method

Abstract

For an arbitrary self-adjoint operator B in a Hilbert space $\mathfrak{H}$, we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector $x \in \mathfrak{H}$ with respect to the operator B, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator B, and the k-modulus of continuity of the vector x with respect to the operator B. The results are used for finding a priori estimates for the Ritz approximate solutions of operator equations in a Hilbert space.