Related papers: Approximation of $L^\infty$ functionals with gener…

In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will…

In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the…

We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:\Omega\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the…

We propose an abstract framework for the homogenization of random functionals which may contain non-convex terms, based on a two-scale $\Gamma$-convergence approach and a definition of Young measures on micropatterns which encodes the…

Under various conditions on the data we analyse how appearence of lower order terms affects the gradient estimates on solutions to a general nonlinear elliptic equation of the form \[-{\rm div}\, a(x,Du)+b(x,u)=\mu\] with data $\mu$ not…

This note completely resolves the asymptotic development of order $2$ by $\Gamma$-convergence of the mass-constrained Cahn--Hilliard functional, by showing that one of the critical assumptions of the authors' previous work (Leoni, Murray,…

In this work, we study convergence in probability and almost sure convergence for weighted partial sums of random variables that are related to the class of generalized Oppenheim expansions. It is worth noting that the random variables…

In this paper we derive some asymptotic formulas for the $q$-Gamma function $\Gamma_{q}(z)$ for $q$ tending to 1.

In a decision-theoretic framework, the minimax lower bound provides the worst-case performance of estimators relative to a given class of statistical models. For parametric and semiparametric models, the H\'{a}jek--Le Cam local asymptotic…

We study the central limit theorem in the non-normal domain of attraction to symmetric $\alpha$-stable laws for $0<\alpha\leq2$. We show that for i.i.d. random variables $X_i$, the convergence rate in $L^\infty$ of both the densities and…

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We…

The $\Gamma $-limit of a family of functionals $u\mapsto \int_{\Omega }f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx$ is obtained for $s=1,2$ and when the integrand $f=f\left( y,z,v\right) $ is a continous…

A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some…

In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point…

In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and $\Gamma$-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the…

We provide comprehensive regularity results and optimal conditions for a general class of functionals involving Orlicz multi-phase of the type \begin{align} \label{abst:1} v\mapsto \int_{\Omega} F(x,v,Dv)\,dx, \end{align} exhibiting…

We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics…

Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the…

We study the $\Gamma$-convergence of the functionals $F_n(u):= || f(\cdot,u(\cdot),Du(\cdot))||_{p_n(\cdot)}$ and $\mathcal{F}_n(u):= \int_{\Omega} \frac{1}{p_n(x)} f^{p_n(x)}(x,u(x),Du(x))dx$ defined on $X\in \{L^1(\Omega,\mathbb{R}^d),…

A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to…