en>Vieque: Disambiguated: John Burgess → John P. Burgess

2014-02-04T01:53:23Z

Disambiguated: John Burgess → John P. Burgess

← Older revision		Revision as of 03:53, 4 February 2014
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	~~Their author~~ is ~~known through name~~ of ~~Gabrielle Lattimer though she~~ [~~http~~://~~Search~~.~~Un.org/search?ie~~=~~utf8&site~~=~~un_org&output~~=~~xml_no_dtd&client~~=~~UN_Website_en&num~~=~~10&lr~~=~~lang_en&proxystylesheet~~=~~UN_Website_en&oe~~=~~utf8&q~~=~~doesn%27t+tremendously~~&~~Submit=Go doesn~~'~~t tremendously] like being called in this way. For years she~~'~~s been working because a [http:~~//~~www~~.~~encyclopedia.com~~/~~searchresults.aspx?q=library+assistant library assistant]. To bake~~ is ~~something that jane is been doing~~ for ~~months~~ and ~~months. For years she~~'s ~~been tough to adapt in Massachusetts. She is running and exercising a blog here~~: ~~http~~://~~prometeu~~.~~net~~<br><br>~~Feel free to visit my web site~~ .~~.. clash of clans hack tool (~~[~~http~~:~~//prometeu.net why not try this out~~])		In [[mathematics]], the '''Fejér kernel''' is used to express the effect of [[Cesàro summation]] on [[Fourier series]]. It is a non-negative kernel, giving rise to an [[approximate identity]].
			[[Image:Fejér kernel.svg\|thumb\|400px\|Plot of several Fejér kernels]]
			The '''Fejér kernel''' is defined as

			:<math>F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}D_k(x),</math>

			where
			:<math>D_k(x)=\sum_{s=-k}^k {\rm e}^{isx}</math>
			is the ''k''th order [[Dirichlet kernel]]. It can also be written in a closed form as

			:<math>F_n(x) = \frac{1}{n} \left(\frac{\sin \frac{n x}{2}}{\sin \frac{x}{2}}\right)^2 =
			\frac{1}{n} \frac{1 - \cos(nx)}{1 - \cos x}
			</math>,

			where this expression is defined.<ref>{{cite book \|title=Banach Spaces of Analytic Functions \|last=Hoffman \|first=Kenneth \|year=1988 \|publisher=Dover \|isbn=0-486-45874-1 \|page=17 }}</ref> It is named after the [[Hungary\|Hungarian]] mathematician [[Lipót Fejér]] (1880–1959).

			The important property of the '''Fejér kernel''' is <math>F_n(x) \ge 0</math> with average value of <math>1 </math>. The [[convolution]] ''F<sub>n</sub>'' is positive: for <math>f \ge 0</math> of period <math>2 \pi</math> it satisfies

			:<math>0 \le (f*F_n)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(y) F_n(x-y)\,dy,</math>

			and, by [[Young's inequality]],
			:<math>\\|F_n*f \\|_{L^p([-\pi, \pi])} \le \\|f\\|_{L^p([-\pi, \pi])}</math> for every <math>0 \le p \le \infty</math>
			for continuous function <math>f</math>; moreover,
			:<math>f*F_n \rightarrow f</math> for every <math>f \in L^p([-\pi, \pi])</math> (<math>1 \le p < \infty</math>)
			for [[Continuous function (topology)\|continuous]] function <math>f</math>. Indeed, if <math>f</math> is continuous, then the convergence is uniform.

			==See also==
			* [[Fejér's theorem]]
			* [[Dirichlet kernel]]
			* [[Gibbs phenomenon]]
			* [[Charles Jean de la Vallée-Poussin]]

			==References==
			<references/>

			{{DEFAULTSORT:Fejer Kernel}}
			[[Category:Fourier series]]

en>ChrisGualtieri: /* Definition */TypoScan Project / General Fixes, typos fixed: , → , using AWB

2012-04-27T14:46:51Z

Definition: TypoScan Project / General Fixes, typos fixed: , → , using AWB

New page

Their author is known through name of Gabrielle Lattimer though she [http://Search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=doesn%27t+tremendously&Submit=Go doesn't tremendously] like being called in this way. For years she's been working because a [http://www.encyclopedia.com/searchresults.aspx?q=library+assistant library assistant]. To bake is something that jane is been doing for months and months. For years she's been tough to adapt in Massachusetts. She is running and exercising a blog here: http://prometeu.net<br><br>Feel free to visit my web site ... clash of clans hack tool ([http://prometeu.net why not try this out])

Μ operator - Revision history

en>Vieque: Disambiguated: John Burgess → John P. Burgess

en>ChrisGualtieri: /* Definition */TypoScan Project / General Fixes, typos fixed: , → , using AWB