Sampling in BV-Type Spaces

TL;DR

This paper establishes the theoretical foundations for sampling functions of bounded variation (BV), proving the continuity of sampling functionals and the invertibility of the differential operator, which aids in solving BV-based inverse problems.

Contribution

It introduces a canonical inversion of the differential operator D and provides an existence theorem for BV sampling-based optimization problems.

Findings

Proves the continuity of sampling functionals for BV functions.

Shows the differential operator D has a unique local inverse.

Characterizes the solution set in terms of extreme points.

Abstract

The sampling of functions of bounded variation (BV) is a long-standing problem in op- timization. The ability to sample such functions has relevance in the field of variational inverse problems, where the standard theory fails to guarantee the mere existence of solutions when the loss functional involves samples of BV functions. In this paper, we prove the continuity of sampling functionals and show that the differential operator D admits a unique local inverse. This canonical inversion enables us to formulate an existence theorem for a class of regularized optimization problems that incorporate samples of BV functions. Finally, we characterize the solution set in terms of its extreme points.