On approximation of solutions of operator-differential equations with their entire solutions of exponential type

TL;DR

This paper studies how well solutions of a certain operator-differential equation can be approximated by entire solutions of exponential type, linking approximation quality to the solution's smoothness.

Contribution

It provides direct and inverse approximation theorems for solutions of operator-differential equations using entire exponential type solutions, establishing a smoothness-approximation relationship.

Findings

Established a one-to-one correspondence between approximation convergence rate and solution smoothness.

Provided explicit approximation theorems for solutions involving nonnegative self-adjoint operators.

Illustrated results with an example involving elliptic differential operators in bounded domains.

Abstract

We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0, \infty)$, where $A$ is a nonnegative self-adjoint operator in a Hilbert space. We give direct and inverse theorems on approximation of solutions of this equation with its entire solutions of exponential type. This establishes a one-to-one correspondence between the order of convergence to $0$ of the best approximation of a solution and its smoothness degree. The results are illustrated with an example, where the operator $A$ is generated by a second order elliptic differential expression in the space $L_{2}(\Omega)$ \ (the domain $\Omega \subset \mathbb{R}^{n}$ is bounded with smooth boundary) and a certain boundary condition.