Fuzzy best simultaneous approximation of a finite number of functions is considered. For this purpose, a fuzzy norm on $Cleft (X, Y right )$ and its fuzzy dual space and also the  set of subgradients of a fuzzy norm are introduced. Necessary case of a proved theorem about characterization of simultaneous approximation will be extended to the fuzzy case.

We introduce a novel generic energy functional that we employ to solve inverse imaging problems within a variational framework. The proposed regularization family, termed as structure tensor total variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both grayscale and vector-valued images. It generalizes several existing variational penalties, including the tot...

Abstract Our aim is to characterize the homogeneous fractional Sobolev–Slobodecki? spaces $$\mathcal {D}^{s,p} (\mathbb {R}^n)$$ D s , p ( R n ) ...

It is well known that finite element solutions for elliptic PDEs with Dirac measures as source terms converge, due to the fact that the solution is not in H1, suboptimal in classical norms. A standard remedy is to use graded meshes, then quasioptimality, i.e., optimal up to a log-factor, for low order finite elements can be recovered, e.g., in the L2-norm. Here we show for the lowest order case...

In regularized kernel methods, the solution of a learning problem is found by minimizing functionals consisting of the sum of a data and a complexity term. In this paper we investigate some properties of a more general form of the above functionals in which the data term corresponds to the expected risk. First, we prove a quantitative version of the representer theorem holding for both regressi...

we introduce the notions of t-dual rickart and strongly t-dual rickart modules. we provide several characterizations and investigate properties of each of these concepts. it is shown that every free (resp. finitely generated free) $r$-module is t-dual rickart if and only if $overline{z}^2(r)$ is a  direct summand of $r$ and end$(overline{z}^2(r))$ is a semisimple (resp. regular) ring. it is sho...

We study the topological spectrum of a seminormed ring $R$ which we define as space prime ideals $\mathfrak{p}$ such that equals kernel some bounded power-multiplicative seminorm. For any show is quasi-compact sober space. When perfectoid Tate construct natural homeomorphism between and its tilt $R^{\flat}$. As an application, prove integral domain if only domain.

Let I be a finite interval and r ∈ N. Denote by ∆ s + L q the subset of all functions y ∈ L q such that the s-difference ∆ s τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by ∆ s + W r p , the class of functions x on I with the seminorm x (r) L p ≤ 1, such that ∆ s τ x ≥ 0, τ > 0. For s = 3,. .. , r + 1, we obtain two-sided estimates of the shape preserving widths