Related papers: On weighted polynomial approximation

We give an elementary proof of an analogue of Fej\'er's theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces.

We prove effective results on when a function can be approximated by a Dirichlet polynomial with bounded coefficients. Assuming that \Phi(n) is an increasing function we prove that the set of polynomials {\sum_{n=2}^N a_n n^{it-1}: N \geq…

The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result…

We prove that if $\{ \varphi_j\}_j$ is a sequence of subharmonic functions which are increasing to some subharmonic function $\varphi$ in $\mathbb{C}$, then the union of all the weighted Hilbert spaces $H(\varphi_j)$ is dense in the…

We show that for multivariate Freud-type weights $W_\alpha(x)=\exp(-|x|^\alpha)$, $\alpha>1$, any convex function $f$ on $R^d$ satisfying $fW_\alpha\in L_p(R^d)$ if $1\le p<\infty$, or $\lim_{|x|\to\infty}f(x)W_\alpha(x)=0$ if $p=\infty$,…

We study the density of polynomials in $H^2(\Omega,e^{-\varphi})$, the space of square integrable holomorphic functions in a bounded domain $\Omega$ in $\mathbb{C}$, where $\varphi$ is a subharmonic function. In particular, we prove that…

We formulate and discuss a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure…

Let $S$ be a dense subring of the real numbers. In this paper we prove a polynomial version of Van der Waerden's theorem near zero. In fact, we prove that if $p_1,\ldots,p_m \in \mathbb{Z}[x]$ are polynomials such that $p_i(0) = 0$ and…

This note mainly concerns the binomial power function, defined as $(1+x^q)^{r}$. We construct systems of polynomials related to non-local approximation, which allows us to establish the density results on $C[a,b]$, where $a,b\in\mathbb{R}$.…

Among other things, we prove that, for a doubling weight $w$, $0< p\leq\infty$, $r\in{\mathbb N}_0$, and $0<\alpha <r+1 - 1/\lambda_p$, we have \[ E_n(f)_{p, w_n} = O(n^{-\alpha}) \iff \omega_\varphi^{r+1}(f, n^{-1})_{p, w_n} =…

In this note we study a quantitative version of Bernstein's approximation problem when the polynomials are dense in weighted spaces on the real line completing a result of S.~N.~Mergelyan (1960). We estimate in the logarithmic scale the…

We prove that smooth $C^\infty$ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a~similar result in non-weighted spaces defined by some kernel…

Final representation of all those measures $\mu$ for which algebraic polynomials are dense in $L_p(R, d\mu)$ is found. The weighted analogue of the Weierstrass polynomial approximation theorem and a new version of the M. Krein's theorem…

A classical result in approximation theory states that for any continuous function \( \varphi: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \(…

Consider the Mills ratio $f(x)=\big(1-\Phi(x)\big)/\phi(x), \, x\ge 0$, where $\phi$ is the density function of the standard Gaussian law and $\Phi$ its cumulative distribution.We introduce a general procedure to approximate $f$ on the…

Let $\mathscr{C}_\mathbb{Z}([0,1])$ be the metric space of real-valued continuous functions on $[0,1]$ with integer values at $0$ and $1$, equipped with the uniform (supremum) metric $d_\infty$. It is a classical theorem in approximation…

A space $G(M, \varPhi)$ of infinitely differentiable functions in ${\mathbb R}^n$ constructed with a help of a family $\varPhi=\{\varphi_m\}_{m=1}^{\infty}$ of real-valued functions $\varphi_m \in~C({\mathbb R}^n)$ and a logarithmically…

Let $X$ be a Banach holomorphic function space on the unit disk. A linear polynomial approximation scheme for $X$ is a sequence of bounded linear operators $T_n:X\to X$ with the property that, for each $f\in X$, the functions $T_n(f)$ are…

In this paper, we prove a quantitative approximation result by orthonormal polynomials associated to an exponential weight of the form e -$\Phi$ , where $\Phi$ is an even polynomial with positive leading coefficient. This result is a…

There exist well-known tight bounds on the error between a function $f \in C^{\,n + 1}([-1, 1])$ and its best polynomial approximation of degree $n$. We show that the error meets these bounds when and only when $f$ is a polynomial of degree…