Related papers: Algebraic Versus Analytic Density of Polynomials

We show that under very mild conditions on a measure $\mu$ on the interval $[0,\infty)$, the span of $\{x^k\}_{k=n}^{\infty}$ is dense in $L^2(\mu)$ for any $n=0,1,\ldots$. We present two different proofs of this result, one based on the…

We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real algebraic variety $M\subset\R^n$ is dense in the Hilbert space $L^2(M,e^{-|x|^2}\de\mu)$, where $\de\mu$ denotes the volume form of $M$ and $\de\nu=e^{-|x|^2}\de\mu$…

This paper presents an erroneous proof that if the polynomials are dense in $L_2(\mathbb{R}, \rho)$, then they are dense in $L_2(\mathbb{R}, \rho+\mu)$ where $\mu$ is a measure supported on a finite set of points.

In this paper we study the density of polynomials in some $L^2(M)$ spaces. Two choices of the measure $M$ and polynomials are considered: 1) a $(N\times N)$ matrix non-negative Borel measure on $\mathbb{R}$ and vector-valued polynomials…

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue $L^2$-space to the existence of analytic bounded point evaluations. Analogues to the complex plane…

A classical theorem of Szeg\H{o} states that for any probability measure $\mu=w\frac{\mathrm{d}\theta}{2\pi}+\mu_s$ on the unit circle the polynomials are dense in $L^2(\mathbb{T},\mu)$ if and only if $\log(w)\notin L^1(\mathbb{T})$. A…

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…

It is shown by the author in [J. Lie Theory 29:4, 1045-1070, 2019] that for every connected linear complex Lie group the algebra of polynomials (regular functions) is dense in the algebra of holomorphic functions of exponential type.…

Let $X$ be a separable Banach function space on the unit circle $\mathbb{T}$ and $H[X]$ be the abstract Hardy space built upon $X$. We show that the set of analytic polynomials is dense in $H[X]$ if the Hardy-Littlewood maximal operator is…

We prove that if a Borel probability measure (\mu) on (\T) is invariant under the action of a "large" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then (\mu) is either…

This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by $L^2$.…

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces $L^{2}(\mu)$, with $\mu$ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix to the…

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…

Let $G$ be an LCA group, $H$ a closed subgroup, $\varGamma$ the dual group of $G$ and $\mu$ be a regular finite non-negative Borel measure on $\varGamma$. We give some necessary and sufficient conditions for the density of the set of…

We consider self-similar measures $\mu $ with support in the interval $0\leq x\leq 1$ which have the analytic functions $\left\{e^{i2\pi nx}:n=0,1,2,... \right\} $ span a dense subspace in $L^{2}(\mu) $. Depending on the fractal dimension…

We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem. Let f be a proper holomorphic map from an open set U onto a Stein manifold V, $U\subset\subset V$. Assume f is…

We prove the density of the sets of the form ${{\lambda}_1^m {\mu}_1^n {\xi}_1 +...+{\lambda}_k^m {\mu}_k^n {\xi}_k : m,n \in \mathbb N}$ modulo one, where $\lambda_i$ and $\mu_i$ are multiplicatively independent algebraic numbers…

Let $\{F_n\}$ be the sequence of the Fej\'er kernels on the unit circle $\mathbb{T}$. The first author recently proved that if $X$ is a separable Banach function space on $\mathbb{T}$ such that the Hardy-Littlewood maximal operator $M$ is…

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…

The paper, that continuous some previous work of Sch\"onherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive…