Nonlocal Bounded Variations with Applications
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Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation (BV)-type spaces. Two different natural fractional analogs of classical BV are considered: BV^α, a space induced from the Riesz-fractional gradient that has been recently studied by Comi-Stefani; and bv^α, induced by the Gagliardo-type fractional gradient often used in Dirichlet forms and Peridynamics - this one is naturally related to the Caffarelli-Roquejoffre-Savin fractional perimeter. Our main theoretical result is that the latter bv^α actually corresponds to the Gagliardo-Slobodeckij space W^α,1. As an application, using the properties of these spaces, novel image denoising models are introduced and their corresponding Fenchel pre-dual formulations are derived. The latter requires density of smooth functions with compact support. We establish this density property for convex domains.
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