en>Headbomb: Various citation cleanup (identifiers mostly) using AWB

2011-09-05T07:43:18Z

Various citation cleanup (identifiers mostly) using AWB

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In [[mathematical analysis]], '''Krein's condition''' provides a necessary and sufficient condition for exponential sums

:<math> \left\{ \sum_{k=1}^n a_k \exp(i \lambda_k x),
\quad a_k \in \mathbb{C}, \, \lambda_k \geq 0 \right\},\, </math>

to be [[dense (topology)|dense]] in a [[Lp-space#Weighted Lp spaces|weighted L<sub>2</sub> space]] on the real line. It was discovered by [[Mark Krein]] in the 1940s.<ref>{{cite journal|last=Krein|first=M.G.|author-link=Mark Krein|title=On an extrapolation problem due to Kolmogorov|journal=[[Doklady Akademii Nauk SSSR]]|volume= 46|year=1945|pages=306–309}}</ref> A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the [[moment problem]].<ref>{{eom|id=Krein_condition|first=J.|last=Stoyanov}}</ref><ref>{{cite journal|last=Berg|first=Ch.|title=Indeterminate moment problems and the theory of entire functions|doi=10.1016/0377-0427(95)00099-2|journal=J. Comput. Appl. Math.|volume=65|year=1995|pages=1–3, 27–55|mr=1379118}}</ref>

==Statement==

Let ''μ'' be an [[Absolutely continuous#Absolute continuity of measures|absolutely continuous]] [[measure (mathematics)|measure]] on the real line, d''μ''(''x'') = ''f''(''x'') d''x''. The exponential sums

:<math> \sum_{k=1}^n a_k \exp(i \lambda_k x),
\quad a_k \in \mathbb{C}, \, \lambda_k \geq 0 </math>

are dense in ''L''<sub>2</sub>(''μ'') if and only if

:<math> \int_{-\infty}^\infty \frac{- \ln f(x)}{1 + x^2} \, dx = \infty. </math>

==Indeterminacy of the moment problem==

Let ''μ'' be as above; assume that all the [[moment (mathematics)|moments]]

:<math> m_n = \int_{-\infty}^\infty x^n d\mu(x), \quad n = 0,1,2,\ldots</math>

of ''μ'' are finite. If

:<math> \int_{-\infty}^\infty \frac{- \ln f(x)}{1 + x^2} \, dx < \infty </math>

holds, then the [[Hamburger moment problem]] for ''μ'' is indeterminate; that is, there exists another measure ''ν'' ≠ ''μ'' on '''R''' such that

:<math> m_n = \int_{-\infty}^\infty x^n \, d\nu(x), \quad n = 0,1,2,\ldots</math>

This can be derived from the "only if" part of Krein's theorem above.<ref>{{Cite book |first=N. I. |last=Akhiezer |author-link=Naum Akhiezer|title=The Classical Moment Problem and Some Related Questions in Analysis |location= |publisher=Oliver & Boyd |year=1965 }}</ref>

===Example===

Let

:<math> f(x) = \frac{1}{\sqrt{\pi}} \exp \left\{ - \ln^2 x \right\};</math>

the measure d''μ''(''x'') = ''f''(''x'') d''x'' is called the [[Stieltjes–Wigert polynomials|Stieltjes–Wigert measure]]. Since

:<math>
\int_{-\infty}^\infty \frac{- \ln f(x)}{1+x^2} dx
= \int_{-\infty}^\infty \frac{\ln^2 x + \ln \sqrt{\pi}}{1 + x^2} \, dx < \infty, </math>

the Hamburger moment problem for ''μ'' is indeterminate.

==References==
{{Reflist}}

[[Category:Mathematical analysis]]
[[Category:Theorems in approximation theory]]

Q-Meixner polynomials - Revision history

en>Headbomb: Various citation cleanup (identifiers mostly) using AWB