# Density questions in the classical theory of moments

• Volume: 31, Issue: 3, page 99-114
• ISSN: 0373-0956

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## Abstract

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Let $\mu$ be a positive Radon measure on the real line having moments of all orders. We prove that the set $P$ of polynomials is note dense in ${L}^{p}\left(\mathbf{R},\mu \right)$ for any $p>2$, if $\mu$ is indeterminate. If $\mu$ is determinate, then $P$ is dense in ${L}^{p}\left(\mathbf{R},\mu \right)$ for $1\le p\le 2$, but not necessarily for $p>2$. The compact convex set of positive Radon measures with same moments as $\mu$ is studied in some details.

## How to cite

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Berg, Christian, and Christensen, J. P. Reus. "Density questions in the classical theory of moments." Annales de l'institut Fourier 31.3 (1981): 99-114. <http://eudml.org/doc/74510>.

@article{Berg1981,
abstract = {Let $\mu$ be a positive Radon measure on the real line having moments of all orders. We prove that the set $P$ of polynomials is note dense in $L^p(\{\bf R\},\mu )$ for any $p&gt;2$, if $\mu$ is indeterminate. If $\mu$ is determinate, then $P$ is dense in $L^p(\{\bf R\},\mu )$ for $1\le p \le 2$, but not necessarily for $p&gt;2$. The compact convex set of positive Radon measures with same moments as $\mu$ is studied in some details.},
author = {Berg, Christian, Christensen, J. P. Reus},
journal = {Annales de l'institut Fourier},
keywords = {density questions; classical theory of moments; positive Radon measure},
language = {eng},
number = {3},
pages = {99-114},
publisher = {Association des Annales de l'Institut Fourier},
title = {Density questions in the classical theory of moments},
url = {http://eudml.org/doc/74510},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Berg, Christian
AU - Christensen, J. P. Reus
TI - Density questions in the classical theory of moments
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 3
SP - 99
EP - 114
AB - Let $\mu$ be a positive Radon measure on the real line having moments of all orders. We prove that the set $P$ of polynomials is note dense in $L^p({\bf R},\mu )$ for any $p&gt;2$, if $\mu$ is indeterminate. If $\mu$ is determinate, then $P$ is dense in $L^p({\bf R},\mu )$ for $1\le p \le 2$, but not necessarily for $p&gt;2$. The compact convex set of positive Radon measures with same moments as $\mu$ is studied in some details.
LA - eng
KW - density questions; classical theory of moments; positive Radon measure
UR - http://eudml.org/doc/74510
ER -

## References

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1. [1] N.I. AKHIEZER, The classical moment problem, Oliver and Boyd, Edinburgh, 1965. Zbl0135.33803
2. [2] H. BAUER, Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie, De Gruyter, Berlin, 1978. Zbl0381.60001
3. [3] G. FREUD, Orthogonal polynomials, Pergamon Press, Oxford, 1971. Zbl0226.33014
4. [4] E. HEWITT, Remark on orthonormal sets in L2 (a, b), Amer. Math. Monthly, 61 (1954), 249-250. Zbl0055.06002MR15,631e
5. [5] M.A. NAIMARK, Extremal spectral functions of a symmetric operator, Izv. Akad. Nauk. SSSR, ser. matem., 11 ; Dokl. Akad. Nauk. SSSR, 54 (1946), 7-9. Zbl0061.26006MR8,386g
6. [6] R.R. PHELPS, Lectures on Choquet's Theorem, Van Nostrand, New York, 1966. Zbl0135.36203MR33 #1690
7. [7] M. RIESZ, Sur le problème des moments et le théorème de Parseval correspondant, Acta Litt. Ac. Sci., Szeged., 1 (1923), 209-225. Zbl49.0708.02JFM49.0708.02
8. [8] J.A. SHOHAT and J.D. TAMARKIN, The problem of moments, AMS, New York, 1943. Zbl0063.06973MR5,5c
9. [9] E.M. STEIN and G. WEISS, Introduction to Fourier Analysis on Euclidean spaces, Princeton University Press, 1971. Zbl0232.42007MR46 #4102

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