Some remarks on the stability of the Cauchy equation and completeness

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.

Cauchy difference priors for edge-preserving Bayesian inversion

We consider inverse problems in which the unknown target includes sharp edges, for example interfaces between different materials. Such problems are typical in image reconstruction, tomography, and other inverse problems algorithms. A common solution for edge-preserving inversion is to use total variation (TV) priors. However, as shown by Lassas and Siltanen 2004, TV-prior is not discretization-invariant: the edge-preserving property is lost when the computational mesh is made denser and denser. In this paper we propose another class of priors for edge-preserving Bayesian inversion, the Cauchy difference priors. We construct Cauchy priors starting from continuous one-dimensional Cauchy motion, and show that its discretized version, Cauchy random walk, can be used as a non-Gaussian prior for edge-preserving Bayesian inversion. We generalize the methodology to two-dimensional Cauchy fields, and briefly consider a generalization of the Cauchy priors to Lévy α-stable random field priors. We develop a suitable posterior distribution sampling algorithm for conditional mean estimates with single-component Metropolis–Hastings. We apply the methodology to one-dimensional deconvolution and two-dimensional X-ray tomography problems.