Optimal Convergence Rates for Tikhonov Regularization in Besov Scales

TL;DR

This paper investigates the convergence rates of Tikhonov regularization within Besov space scales, revealing that stronger source conditions can sometimes result in weaker convergence, and establishes optimal source conditions for these spaces.

Contribution

It introduces the analysis of Tikhonov regularization in Besov scales, highlighting differences from Hilbert scales and providing optimal source conditions.

Findings

Regularization in Banach scales can behave differently than in Hilbert scales.

Stronger source conditions may lead to weaker convergence rates.

Optimal source conditions for Besov scales are established.

Abstract

In this paper we deal with linear inverse problems and convergence rates for Tikhonov regularization. We consider regularization in a scale of Banach spaces, namely the scale of Besov spaces. We show that regularization in Banach scales differs from regularization in Hilbert scales in the sense that it is possible that stronger source conditions may lead to weaker convergence rates and vive versa. Moreover, we present optimal source conditions for regularization in Besov scales.