arXiv subjects

Christian Bargetz

Publications and source records attributed to Christian Bargetz.

22 records · Page 2Linked to original sources

arXiv2017

Convergence Properties of Dynamic String Averaging Projection Methods in the Presence of Perturbations

Assuming that the absence of perturbations guarantees weak or strong convergence to a common fixed point, we study the behavior of perturbed products of an infinite family of nonexpansive operators. Our main result indicates that the convergence rate of unperturbed products is essentially preserved in the presence of perturbations. This, in particular, applies to the linear convergence rate of dynamic string averaging projection methods, which we establish here as well. Moreover, we show how this result can be applied to the superiorization methodology.

math.NA↗

arXiv2017

Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(κ)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $σ$-porous subset. For separable metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.

math.MG↗

arXiv2016

σ-Porosity of the set of strict contractions in a space of non-expansive mappings

We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uni- form convergence. We show that the set of strict contractions is a σ-porous subset.

math.FA↗

arXiv2015

Kernel Identities and Vectorial Regularization

We present the method of "vectorial regularization" to prove kernel identities. This method is applied to derive both known kernel identities, e.g. $\dot{\mathcal{B}}_{xy}=\dot{\mathcal{B}}_x\widehat{\otimes}_\varepsilon\dot{\mathcal{B}}_y$, $\mathcal{D}'_{L^1,xy}=\mathcal{D}'_{L^1,x}\widehat{\otimes}_π\mathcal{D}'_{L^1,y}$, as well as new ones: $\dot{\mathcal{B}}'_{xy}=\dot{\mathcal{B}}'_x\widehat{\otimes}_\varepsilon\dot{\mathcal{B}}'_y$ and $\mathcal{D}_{L^1,xy}=\mathcal{D}_{L^1,x}\widehat{\otimes}_π\mathcal{D}_{L^1,y}$.

math.FA↗