About a deficit in low-order convergence rates on the example of autoconvolution

Abstract

We revisit in -spaces the autoconvolution equation with solutions which are real-valued or complex-valued functions defined on a finite real interval, say . Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity, deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low-order convergence rates for regularized solutions of nonlinear ill-posed operator equations with solutions in a Hilbert space setting. So, for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex -functions with a fixed and uniformly bounded modulus function.

Keywords:

• autoconvolution equation
• inverse problems
• local well-posedness and ill-posedness
• Tikhonov regularization
• source conditions
• convergence rates

Classification:

• 65J20
• 45G10
• 47J06
• 47A52
• 65J15

Acknowledgments

The authors express their thanks to Qinian Jin (Australian National University Canberra), Barbara Kaltenbacher (Univ. Klagenfurt) and Jens Flemming (TU Chemnitz) for helpful discussions and valuable hints. SB and BH greatly appreciate financial support by the German Research Foundation (DFG) under grants FL 832/1-1 and HO 1454/9-1, respectively.