Range inclusions and approximate source conditions with general benchmark functions

The paper is devoted to the analysis of ill-posed operator equations Ax = y with injective linear operator A and solution x0 in a Hilbert space setting. We present some new ideas and results for finding convergence rates in Tikhonov regularization based on the concept of approximate source conditions by means of using distance functions with a general benchmark. For the case of compact operator A and bench-mark functions of power-type we can show that there is a one-to-one correspondence between the maximal power-type decay rate of the distance function and the best possible Hölder exponent for the noise-free convergence rate in Tikhonov regular-ization. As is well-known this exponent coincides with the supremum of exponents in power-type source conditions. The main theorem of this paper is devoted to the impact of range inclusions under the smoothness assumption that x0 is in the range of some positive self-adjoint operator G. It generalizes a convergence rate result proven for compact G in [12] to the case of general operators G with non-closed range