# Bernstein’s Problem on Weighted Polynomial Approximation

@article{Poltoratski2015BernsteinsPO, title={Bernstein’s Problem on Weighted Polynomial Approximation}, author={Alexei Poltoratski}, journal={arXiv: Complex Variables}, year={2015}, pages={147-171} }

We formulate 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 μ on the real line we give a criterion for density of polynomials in L p (μ).

## 9 Citations

On a Uniqueness Theorem of E. B. Vul

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We recall a uniqueness theorem of E. B. Vul pertaining to a version of the cosine transform originating in spectral theory. Then we point out an application to the Bernstein approximation problem…

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We prove some uniqueness problems for meromorphic inner functions on the upper half-plane. In these problems we consider spectral data depending partially or fully on the spectrum, derivative values…

De Branges’ theorem on approximation problems of Bernstein type

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Abstract The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted ${C}_{0} $-space on the real line. A theorem of de Branges…

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We study the localization of zeros of Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular,…

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Mathematical shapes of uncertainty Gap theorems A problem by Polya and Levinson Determinacy of measures and oscillations of high-pass signals Beurling-Malliavin and Bernstein's problems The type…

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The goal of this paper is to combine ideas from the theory of mixed spectral problems for differential operators with new results in the area of the Uncertainty Principle in Harmonic Analysis (UP).…

Smoothing of weights in the Bernstein approximation problem

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In 1924 S.Bernstein asked for conditions on a uniformly bounded on $\mathbb{R}$ Borel function (weight) $w: \mathbb{R} \to [0, +\infty )$ which imply the denseness of algebraic polynomials…

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