Sampling in BV-Type Spaces

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.