Boundedness and Unboundedness in Total Variation Regularization

We consider whether minimizers for total variation regularization of linear inverse problems belong to $L^\infty$ even if the measured data does not. present a simple proof boundedness minimizer fixed parameter, and derive existence uniform bounds sufficiently small noise under source condition adequate priori parameter choices. To show that such result cannot be expected every fidelity term dimension we compute an explicit radial unbounded minimizer, which is accomplished by proving equivalence weighted one-dimensional denoising with generalized taut string problem. Finally, discuss possibility extending results related higher-order functionals, obtaining positive answer infimal convolution first second order variation.