en>David Eppstein: link Claude LeBrun

2013-01-31T07:19:59Z

← Older revision		Revision as of 09:19, 31 January 2013
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	I'~~m Yoshiko Oquendo~~. I~~'ve usually cherished residing in Idaho~~. ~~One~~ of ~~my favorite hobbies~~ is ~~tenting and now I'm attempting to make cash with~~ it. ~~Bookkeeping~~ is ~~what I do~~ for a ~~living.~~<br><br>~~Feel free to surf to my web page~~: ~~extended auto warranty~~; [~~http~~://~~www~~.~~Dienoobs~~-~~Css~~.~~de/index~~.~~php?mod~~=~~users&action~~=~~view&~~id=~~17012 Read Homepage~~],		In [[approximation theory]], '''Jackson's inequality''' is an inequality bounding the value of function's best approximation by [[polynomials\|algebraic]] or [[trigonometric polynomials]] in terms of the [[modulus of continuity]] of its derivatives.<ref>{{cite book\|last=Achieser\|first=N.I.\|author-link=Naum Akhiezer\|title=Theory of Approximation\|year=1956\|publisher=Frederick Ungar Publishing Co\|location=New York}}</ref> Informally speaking, the smoother the function is, the better it can be approximated by polynomials.

			==Statement: trigonometric polynomials==

			For trigonometric polynomials, the following was proved by [[Dunham Jackson]]:

			'''Theorem 1''': If ''&fnof;'': [0, 2{{pi}}] → '''C''' is an ''r'' times differentiable [[periodic function]] such that

			: <math>\|f^{(r)}(x)\| \leq 1, \quad 0 \leq x \leq 2\pi,</math>

			then, for every natural ''n'', there exists a [[trigonometric polynomial]] ''P''<sub>''n''−1</sub> of degree at most ''n'' − 1 such that

			: <math>\|f(x) - P_{n-1}(x)\| \leq \frac{C(r)}{n^r}, \quad 0 \leq x \leq 2\pi, </math>

			where ''C''(''r'') depends only on ''r''.

			The '''[[Naum Akhiezer\|Akhiezer]]–[[Mark Krein\|Krein]]–[[Jean Favard\|Favard]] theorem''' gives the sharp value of ''C''(''r'') (called the [[Favard constant\|Akhiezer–Krein–Favard constant]]):

			: <math> C(r) = \frac{4}{\pi} \sum_{k=0}^\infty \frac{(-1)^{k(r+1)}}{(2k+1)^{r+1}}~.</math>

			Jackson also proved the following generalisation of Theorem 1:

			'''Theorem 2''': Denote by ''ω''(''δ'', ''&fnof;''<sup>(''r'')</sup>) the modulus of continuity of the ''r''th derivative of ''&fnof;''. Then one can find ''P''<sub>''n''−1</sub> such that

			: <math>\|f(x) - P_{n-1}(x)\| \leq \frac{C_1(r) \omega(1/n, f^{(r)})}{n^r}, \quad 0 \leq x \leq 2\pi </math>

			==Further remarks==

			Generalisations and extensions are called Jackson-type theorems. A converse to Jackson's inequality is given by [[Bernstein's theorem (approximation theory)\|Bernstein's theorem]]. See also [[constructive function theory]].

			==References==
			{{Reflist}}

			==External links==
			* {{SpringerEOM\|id=Jackson_inequality\|first1=N.P.\|last1=Korneichuk\|first2=V.P.\|last2=Motornyi}}
			* {{MathWorld\|title=Jackson's Theorem\|id=JacksonsTheorem}}

			{{DEFAULTSORT:Jackson's Inequality}}
			[[Category:Approximation theory]]
			[[Category:Inequalities]]
			[[Category:Theorems in approximation theory]]


			{{mathanalysis-stub}}

en>Phuzion: clean up, added orphan tag using AWB

2012-05-11T00:36:00Z

clean up, added orphan tag using AWB

New page

I'm Yoshiko Oquendo. I've usually cherished residing in Idaho. One of my favorite hobbies is tenting and now I'm attempting to make cash with it. Bookkeeping is what I do for a living.<br><br>Feel free to surf to my web page: extended auto warranty; [http://www.Dienoobs-Css.de/index.php?mod=users&action=view&id=17012 Read Homepage],

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en>David Eppstein: link Claude LeBrun

en>Phuzion: clean up, added orphan tag using AWB