Related papers: Bernstein-Walsh inequalities in higher dimensions …
We study the asymptotic behaviour of the approximation, Gelfand and Kolmogorov numbers of the compact embeddings of weighted function spaces of Besov and Triebel-Lizorkin type in the case where the weights belong to a large class. We obtain…
Let A_N be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function D_N(x) in two dimensions with respect to rectangles with lower left corner anchored at the origin and upper right corner at…
We derive the non-asymptotical non-uniform sharp error estimation for Bernstein's approximation of continuous function based on the modern probabilistic apparatus. We investigate also the convergence of derivative of these polynomials and…
Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n…
For the sequence of multi-indexes $\{\alpha_i\}_{i=1}^{m}$ and $\beta$ we study the inequality \[ \|D^{\beta} f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{\alpha_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial…
We consider harmonic functions of polynomial growth of some order $d$ on Cayley graphs of groups of polynomial volume growth of order $D$ w.r.t. the word metric and prove the optimal estimate for the dimension of the space of such harmonic…
Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces. We quantify this fact by showing that a ball of…
We consider expansions of Presburger arithmetic with families of monadic polynomial predicates. (Examples of such predicates are the set of perfect squares, or the set of integers of the form $2n^3-5n+3$, etc.) Although the full attendant…
We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space) of the solution of the linear scalar stochastic delay equation dX(t) = X(t-1) dW(t) which does not depend on the initial…
The following analog of Bernstein inequality for monotone rational functions is established: if $R$ is an increasing on $[-1,1]$ rational function of degree $n$, then $$ R'(x)<\frac{9^n}{1-x^2}\|R\|,\quad x\in (-1,1). $$ The exponential…
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability…
Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such…
We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\|pf-1\|_{\alpha}$ for a given function $f$. For $\alpha\in [0,1]$ (which includes the Hardy and…
We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of given…
We establish pointwise growth estimates for the spherical derivative of solutions of the first order algebraic differential equations. A generalization of this result to higher order equations is also given. We discuss the related question…
A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen \cite{AHH}, \cite {HK-2} that a connection exists between discrete Painlev\'e equations and (finite order) Nevanlinna theory is an estimate of the…
Exponential divided differences arise in numerical linear algebra, matrix-function evaluation, and quantum Monte Carlo simulations, where they serve as kernel weights for time evolution and observable estimation. Efficient and numerically…
Let $f$ be a real function defined on the interval $[0,1]$ which is constant on $(a,b)\subset [0,1]$, and let $B_nf$ be its associated $n$th Bernstein polynomial. We prove that, for any $x\in (a,b)$, $|B_nf(x)-f(x)|$ converges to $0$ as…
Exponential growth occurs when the growth rate of a given quantity is proportional to the quantity's current value. Surprisingly, when exponential growth data is plotted as a simple histogram disregarding the time dimension, a remarkable…
The first aim of this paper is to prove a Gr\"uss-Voronovskaya estimate for Bernstein and for a class of Bernstein-Durrmeyer polynomials on $[0, 1]$. Then, Gr\"uss and Gr\"uss-Voronovskaya estimates for their corresponding operators of…