On the analysis of distance functions for linear ill-posed problems with an application to the integration operator in L²

The paper is devoted to the analysis of linear ill-posed operator equations Ax = y with solution x₀ in a Hilbert space setting. In an introductory part, we recall assertions on convergence rates based on general source conditions for wide classes of linear regularization methods. The source conditions are formulated by using index functions. Error estimates for the regularization methods are developed by exploiting the concept of Mathé and Pereverzev that assumes the qualification of such a method to be an index function. In the main part of the paper we show that convergence rates can also be obtained based on distance functions d(R) depending on radius R > 0 and expressing for x₀ the violation of a benchmark source condition. This paper is focused on the moderate source condition x₀ = A^∗v. The case of distance functions with power type decay rates d(R) = (R^{–η/(1–η)}) as R → ∞ for exponents 0 < η < 1 is especially discussed. For the integration operator in L²(0, 1) aimed at finding the antiderivative of a square-integrable function the distance function can be verified in a rather explicit way by using the Lagrange multiplier method and by solving the occurring Fredholm integral equations of the second kind. The developed theory is illustrated by an example, where the optimal decay order of d(R) → 0 for some specific solution x₀ can be derived directly from explicit solutions of associated integral equations.