# 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.

Published

25.05.2005

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 57, no. 5, May 2005, pp. 633–643, https://umj.imath.kiev.ua/index.php/umj/article/view/3629.

Issue

Section

Research articles