en>BD2412: minor fixes, mostly disambig links using AWB

2013-06-01T17:09:46Z

minor fixes, mostly disambig links using AWB

← Older revision		Revision as of 19:09, 1 June 2013
Line 1:		Line 1:
	~~Friends call him Royal Cummins~~. ~~Her family lives~~ in ~~Delaware but she needs to move because~~ of ~~her family members~~. ~~One~~ of the ~~issues I adore most~~ is ~~climbing~~ and ~~now I have time~~ to ~~consider on new things~~. ~~She functions as a financial officer and she will~~ not ~~change it anytime quickly~~.<br><br>~~Here is my website~~ ... ~~[http:~~//~~unnormal~~.~~clan~~.lc/~~index~~.~~php?mod~~=~~users~~&~~action~~=~~view&id~~=~~1612 clan.lc~~]		In [[mathematical analysis]], the '''Chebyshev–Markov–Stieltjes''' [[inequality (mathematics)\|inequalities]] are inequalities related to the [[moment problem\|problem of moments]] that were formulated in the 1880s by [[Pafnuty Chebyshev]] and proved independently by [[Andrey Markov]] and (somewhat later) by [[Thomas Jan Stieltjes]].<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> Informally, they provide sharp bounds on a [[measure (mathematics)\|measure]] from above and from below in terms of its first [[moment (mathematics)\|moments]].

			==Formulation==

			Given ''m''<sub>0</sub>,...,''m''<sub>2''m''-1</sub> ∈ '''R''', consider the collection '''C''' of measures ''μ'' on '''R''' such that

			: <math>\int x^k d\mu(x) = m_k</math>

			for ''k'' = 0,1,...,2''m'' − 1 (and in particular the integral is defined and finite).

			Let ''P''<sub>0</sub>,''P''<sub>1</sub>, ...,''P''<sub>''m''</sub> be the first ''m'' + 1 [[orthogonal polynomials]] with respect to ''μ'' ∈ '''C''', and let ''ξ''<sub>1</sub>,...''ξ''<sub>''m''</sub> be the zeros of ''P''<sub>''m''</sub>. It is not hard to see that the polynomials ''P''<sub>0</sub>,''P''<sub>1</sub>, ...,''P''<sub>''m''-1</sub> and the numbers ''ξ''<sub>1</sub>,...''ξ''<sub>''m''</sub> are the same for every ''μ'' ∈ '''C''', and therefore are determined uniquely by ''m''<sub>0</sub>,...,''m''<sub>2''m''-1</sub>.

			Denote

			:<math>\rho_{m-1}(z) = 1 \Big/ \sum_{k=0}^{m-1} \|P_k(z)\|^2</math>.

			'''Theorem''' For ''j'' = 1,2,...,''m'', and any ''μ'' ∈ '''C''',

			:<math>\mu(-\infty, \xi_j] \leq \rho_{m-1}(\xi_1) + \cdots + \rho_{m-1}(\xi_j) \leq \mu(-\infty,\xi_{j+1}).</math>

			==References==
			{{Reflist}}

			{{DEFAULTSORT:Chebyshev-Markov-Stieltjes inequalities}}
			[[Category:Mathematical analysis]]
			[[Category:Inequalities]]

en>Michael Hardy at 15:43, 11 March 2012

2012-03-11T15:43:44Z

New page

Friends call him Royal Cummins. Her family lives in Delaware but she needs to move because of her family members. One of the issues I adore most is climbing and now I have time to consider on new things. She functions as a financial officer and she will not change it anytime quickly.<br><br>Here is my website ... [http://unnormal.clan.lc/index.php?mod=users&action=view&id=1612 clan.lc]

Stringology - Revision history

en>BD2412: minor fixes, mostly disambig links using AWB

en>Michael Hardy at 15:43, 11 March 2012