Related papers: Asymptotic Weighted Approximation of Convex Functi…
For a simple model of shallow and wide neural networks, we show that the epigraph of its input-output map as a function of the network parameters approximates epigraph of a. convex function in a precise sense. This leads to a plausible…
We provide asymptotic results for the distribution of weighted nonlinear functionals of Gaussian field with long-range dependence. We also show that integral functionals and the corresponding additive functionals have same distributions…
We derive the dominant asymptotic form and the order of the correction terms of the finite-perimeter partition function of self-avoiding polygons on the square lattice, which are weighted according to their area A as q^A, in the inflated…
We consider a bivariate rational generating function F(x,y) = P(x,y) / Q(x,y) = sum_{r, s} a_{r,s} x^r y^s under the assumption that the complex algebraic curve $\sing$ on which $Q$ vanishes is smooth. Formulae for the asymptotics of the…
It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…
Functions in a Sobolev space are approximated directly by piecewise affine interpolation in the norm of the space. The proof is based on estimates for interpolations and does not rely on the density of smooth functions.
We provide several asymptotic expansions of the prime counting function $\pi(x)$ and related functions. We define an {\it asymptotic continued fraction expansion} of a complex-valued function of a real or complex variable to be a possibly…
We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to…
In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract…
The paper studies the asymptotic behaviour of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given…
We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to $0^+$. We show that dyadic expansions are…
In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional…
Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian…
We propose an algorithm for computing the proximity operator of a sum of composite convex functions in Hilbert spaces and investigate its asymptotic behavior. Applications to best approximation and image recovery are described.
The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet…
We obtain matching direct and inverse theorems for the degree of weighted $L_p$-approximation by polynomials with the Jacobi weights $(1-x)^\alpha (1+x)^\beta$. Combined, the estimates yield a constructive characterization of various…
The paper is devoted to the study of asymptotic behavior of solutions for nonlocal elliptic problems in weighted spaces. We deal with the most difficult case where the support of nonlocal terms intersects with the boundary of a plane…
This article is focused on the asymptotic expansions, as time tends to infinity, of solutions of a system of ordinary differential equations with non-smooth nonlinear terms. The forcing function decays to zero in a very complicated but…
We study the long-time behavior of spatially periodic solutions of the Navier-Stokes equations in the three-dimensional space. The body force is assumed to possess an asymptotic expansion or, resp., finite asymptotic approximation, in…
Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we…