Direct and inverse theorems in the theory of approximation by the Ritz method

TL;DR

This paper establishes direct and inverse approximation theorems linking the smoothness of vectors in a Hilbert space to their approximation rates by exponential-type vectors of a self-adjoint operator, aiding in a priori estimates for Ritz solutions.

Contribution

It introduces new direct and inverse theorems connecting smoothness, approximation rates, and modulus of continuity for self-adjoint operators in Hilbert spaces.

Findings

Established relationships between smoothness and approximation rates.

Derived a priori estimates for Ritz method solutions.

Linked modulus of continuity to approximation convergence.

Abstract

For an arbitrary self-adjoint operator $B$ in a Hilbert space $H$, we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector $x \in H$ 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.