Reassignment method - Revision history

en>Fgnievinski: /* The spectrogram as a time-frequency representation */

2014-11-14T12:40:11Z

The spectrogram as a time-frequency representation

← Older revision		Revision as of 14:40, 14 November 2014
Line 1:		Line 1:
	~~Alyson~~ is ~~what~~ my ~~spouse enjoys to call me~~ but I ~~don't like when people use my full name~~. ~~To perform lacross~~ is ~~something~~ he ~~would never give up~~. ~~Distributing production~~ is ~~exactly where her primary income arrives from~~. ~~For many years she's been residing~~ in ~~Kentucky but her husband wants them to transfer~~.<br><br>my ~~homepage ... psychic chat online~~ - [http://~~mybrandcp~~.com/xe/~~board_XmDx25~~/~~107997 just click the next document~~] -		Royal Votaw is my name but I never truly favored that title. The favorite pastime for him and his children is to generate and now he is attempting to earn money with it. My home is now in Kansas. Managing people is what I do in my working day occupation.<br><br>Here is my web-site; [http://Www.newsfortrader.com/ActivityFeed/MyProfile/tabid/60/userId/29712/language/it-IT/Default.aspx Www.newsfortrader.com]

en>Monkbot: /* The method of reassignment */Fix CS1 deprecated date parameter errors (test)

2014-02-26T04:24:34Z

The method of reassignment: Fix CS1 deprecated date parameter errors (test)

← Older revision		Revision as of 06:24, 26 February 2014
Line 1:		Line 1:
	~~{{distinguish\|Fatou's lemma}}~~		Alyson is what my spouse enjoys to call me but I don't like when people use my full name. To perform lacross is something he would never give up. Distributing production is exactly where her primary income arrives from. For many years she's been residing in Kentucky but her husband wants them to transfer.<br><br>my homepage ... psychic chat online - [http://mybrandcp.com/xe/board_XmDx25/107997 just click the next document] -

	~~In [[complex analysis]], '''Fatou's theorem''', named after [[Pierre Fatou]],~~ is ~~a statement concerning [[holomorphic functions]] on the unit disk and their pointwise extension to the boundary of the disk.~~

	~~==Motivation and statement of theorem==~~

	~~If we have a holomorphic function <math>f</math> defined on the open unit disk <math>D^{2}=\{z:\|z\|<1\}</math>, it is reasonable to ask under~~ what ~~conditions we can extend this function~~ to ~~the boundary of the unit disk. To do this, we can look at what the function looks~~ like on each circle inside the disk centered at 0, each with some radius <math>r</math>. This defines a new function on the circle <math>f_{r}:S^{1}\rightarrow \mathbb{C}</math>, defined by <math>f_{r}(e^{i\theta})=f(re^{i\theta})</math>, where <math>S^{1}:=\{e^{i\theta}:\theta\in[0,2\pi]\}=\{z\in \mathbb{C}:\|z\|=1\}</math>. Then it would be expected that the values of the extension of <math>f</math> onto the circle should be the limit of these functions, and so the question reduces to determining when ~~<math>f_{r}</math> converges, and in what sense, as <math>r\rightarrow 1</math>, and how well defined is this limit~~. ~~In particular, if the [[L-p-space\|<math>L^p</math> norms]] of these <math>f_{r}</math> are well behaved, we have an answer:~~

	~~:'''Theorem:''' Let <math>f:D^{2}\rightarrow\mathbb{C}</math> be a holomorphic function such that~~

	~~:: <math>\sup_{0<r<1}\lVert f_{r}\rVert_{L^{p}(S^{1})}<\infty.</math>~~

	~~Then <math>f_{r}</math> converges to some function <math>f_{1}\in L^{p}(S^{1})</math> [[pointwise]] [[almost everywhere]] and in <math>L^{p}</math>. That~~ is,

	~~:: <math> \lVert f_{r}-f_{1}\rVert_{L^{p}(S^{1})}\rightarrow 0</math>~~

	~~:and~~

	~~:: <math> \|f_{r}(e^{i\theta})-f_{1}(e^{i\theta})\|\rightarrow 0</math>~~
	~~:for almost every <math>\theta\in [0,2\pi]</math>~~.

	~~Now, notice that this pointwise limit~~ is ~~a radial limit. That is, the limit being taken is along a straight line~~ from ~~the center of the disk to the boundary of the circle, and the statement above hence says that~~
	~~:<math> f(re^{i\theta})\rightarrow f_{1}(e^{i\theta})</math>~~
	~~for almost every <math>\theta</math>~~. ~~The natural question is, now with this boundary function defined, will we converge pointwise to this function by taking a limit~~ in ~~any other way? That is, suppose instead of following a straight line~~ to ~~the boundary, we follow an arbitrary curve <math>\gamma:[0,1)\rightarrow D^{2}</math> converging to some point <math>e^{i\theta}</math> on the boundary~~. ~~Will~~ <~~math~~>f<~~/math~~> ~~converge to <math>f_{1}(e^{i\theta})</math>? (Note that the above theorem is just the special case of <math>\gamma(t)=te^{i\theta}</math>)~~.
	It turns out that the curve <math>\gamma</math> needs to be ''nontangential'', meaning that the curve does not approach its target on the boundary in a way that makes it tangent to the boundary of the circle. ~~In other words, the range of <math>\gamma</math> must be contained in a wedge emanating from the limit point~~. ~~We summarize as follows:~~

	~~:'''Definition:''' Let <math>\gamma:[0,1)\rightarrow D^{2}</math> be a continuous path such that <math>\lim_{t\rightarrow 1}\gamma(t)=e^{i\theta}\in S^{1}</math>. Define~~

	~~:: <math>\Gamma_{\alpha}=\{z:\arg z\in~~ [~~\pi-\alpha,\pi+\alpha]\}</math>~~

	~~:and~~

	:: ~~<math>\Gamma_{\alpha}(\theta)=D^{2}\cap e^{i\theta}(\Gamma_{\alpha}+1).<~~/~~math>~~

	~~: That is, <math>\Gamma_{\alpha}(\theta)<~~/~~math> is the wedge inside the disk with angle <math>2\alpha</math> : whose axis passes between <math>e^{i\theta}</math> and zero~~. ~~We say that <math>\gamma<~~/~~math>~~
	~~: converges ''nontangentially'' to <math>e^{i\theta}<~~/~~math>, or that it is a ''nontangential limit'', : if there exists <math>\alpha\in(0,\frac{\pi}{2})<~~/~~math> such that <math>\gamma</math> is contained in <math>\Gamma_{\alpha}</math> and <math>\lim_{t\rightarrow 1}\gamma(t)=e^{i\theta}</math>.~~

	~~:'''Fatou's theorem:''' Let <math>f\in H^{p}(D^{2})</math>. Then for almost all <math>\theta\in[0,2\pi]</math>, <math>\lim_{t\rightarrow1}f(\gamma(t))=f_{1}(e^{i\theta})</math>~~
	~~: for every nontangential limit <math>\gamma</math> converging to <math>e^{i\theta}</math>, where <math>f_{1}</math> is defined as above.~~

	~~==Discussion==~~
	* The proof utilizes the symmetry of the [[Poisson kernel]] using the [[Hardy–Littlewood maximal function]] for the circle.
	* The analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the ~~same way.~~

	~~==See also==~~
	*[[Hardy space]]

	~~==References==~~
	* John B. Garnett, ''Bounded Analytic Functions'', (2006) Springer-~~Verlag, New York~~
	* Walter Rudin. ''Real and Complex Analysis'' (1987), 3rd Ed., McGraw Hill, New York.
	* [[Elias Stein]], ''Singular integrals and differentiability properties of functions'' (1970), Princeton University Press, Princeton.

	~~[[Category:Theorems in complex analysis]]~~

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-29T04:14:36Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 06:14, 29 January 2014
Line 1:		Line 1:
	~~Nice~~ to ~~satisfy you~~, ~~my title~~ is ~~Ling~~ and ~~I completely dig~~ that ~~title~~. ~~Some time~~ in the ~~past he selected~~ to ~~live~~ in ~~Idaho~~. ~~The thing she adores most~~ is to ~~perform handball but she can~~'~~t make~~ it ~~her profession~~. ~~The job he~~'~~s been occupying for years is~~ a ~~messenger~~.<br><br>~~Visit my web blog~~ :: ~~extended car warranty~~ (~~[http~~://~~Sticker-Shocked~~.~~com~~/~~UserProfile~~/~~tabid~~/43/~~userId~~/~~404~~/~~Default~~.~~aspx linked web page~~])		{{distinguish\|Fatou's lemma}}

			In [[complex analysis]], '''Fatou's theorem''', named after [[Pierre Fatou]], is a statement concerning [[holomorphic functions]] on the unit disk and their pointwise extension to the boundary of the disk.

			==Motivation and statement of theorem==

			If we have a holomorphic function <math>f</math> defined on the open unit disk <math>D^{2}=\{z:\|z\|<1\}</math>, it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius <math>r</math>. This defines a new function on the circle <math>f_{r}:S^{1}\rightarrow \mathbb{C}</math>, defined by <math>f_{r}(e^{i\theta})=f(re^{i\theta})</math>, where <math>S^{1}:=\{e^{i\theta}:\theta\in[0,2\pi]\}=\{z\in \mathbb{C}:\|z\|=1\}</math>. Then it would be expected that the values of the extension of <math>f</math> onto the circle should be the limit of these functions, and so the question reduces to determining when <math>f_{r}</math> converges, and in what sense, as <math>r\rightarrow 1</math>, and how well defined is this limit. In particular, if the [[L-p-space\|<math>L^p</math> norms]] of these <math>f_{r}</math> are well behaved, we have an answer:

			:'''Theorem:''' Let <math>f:D^{2}\rightarrow\mathbb{C}</math> be a holomorphic function such that

			:: <math>\sup_{0<r<1}\lVert f_{r}\rVert_{L^{p}(S^{1})}<\infty.</math>

			Then <math>f_{r}</math> converges to some function <math>f_{1}\in L^{p}(S^{1})</math> [[pointwise]] [[almost everywhere]] and in <math>L^{p}</math>. That is,

			:: <math> \lVert f_{r}-f_{1}\rVert_{L^{p}(S^{1})}\rightarrow 0</math>

			:and

			:: <math> \|f_{r}(e^{i\theta})-f_{1}(e^{i\theta})\|\rightarrow 0</math>
			:for almost every <math>\theta\in [0,2\pi]</math>.

			Now, notice that this pointwise limit is a radial limit. That is, the limit being taken is along a straight line from the center of the disk to the boundary of the circle, and the statement above hence says that
			:<math> f(re^{i\theta})\rightarrow f_{1}(e^{i\theta})</math>
			for almost every <math>\theta</math>. The natural question is, now with this boundary function defined, will we converge pointwise to this function by taking a limit in any other way? That is, suppose instead of following a straight line to the boundary, we follow an arbitrary curve <math>\gamma:[0,1)\rightarrow D^{2}</math> converging to some point <math>e^{i\theta}</math> on the boundary. Will <math>f</math> converge to <math>f_{1}(e^{i\theta})</math>? (Note that the above theorem is just the special case of <math>\gamma(t)=te^{i\theta}</math>).
			It turns out that the curve <math>\gamma</math> needs to be ''nontangential'', meaning that the curve does not approach its target on the boundary in a way that makes it tangent to the boundary of the circle. In other words, the range of <math>\gamma</math> must be contained in a wedge emanating from the limit point. We summarize as follows:

			:'''Definition:''' Let <math>\gamma:[0,1)\rightarrow D^{2}</math> be a continuous path such that <math>\lim_{t\rightarrow 1}\gamma(t)=e^{i\theta}\in S^{1}</math>. Define

			:: <math>\Gamma_{\alpha}=\{z:\arg z\in [\pi-\alpha,\pi+\alpha]\}</math>

			:and

			:: <math>\Gamma_{\alpha}(\theta)=D^{2}\cap e^{i\theta}(\Gamma_{\alpha}+1).</math>

			: That is, <math>\Gamma_{\alpha}(\theta)</math> is the wedge inside the disk with angle <math>2\alpha</math> : whose axis passes between <math>e^{i\theta}</math> and zero. We say that <math>\gamma</math>
			: converges ''nontangentially'' to <math>e^{i\theta}</math>, or that it is a ''nontangential limit'', : if there exists <math>\alpha\in(0,\frac{\pi}{2})</math> such that <math>\gamma</math> is contained in <math>\Gamma_{\alpha}</math> and <math>\lim_{t\rightarrow 1}\gamma(t)=e^{i\theta}</math>.

			:'''Fatou's theorem:''' Let <math>f\in H^{p}(D^{2})</math>. Then for almost all <math>\theta\in[0,2\pi]</math>, <math>\lim_{t\rightarrow1}f(\gamma(t))=f_{1}(e^{i\theta})</math>
			: for every nontangential limit <math>\gamma</math> converging to <math>e^{i\theta}</math>, where <math>f_{1}</math> is defined as above.

			==Discussion==
			* The proof utilizes the symmetry of the [[Poisson kernel]] using the [[Hardy–Littlewood maximal function]] for the circle.
			* The analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the same way.

			==See also==
			*[[Hardy space]]

			==References==
			* John B. Garnett, ''Bounded Analytic Functions'', (2006) Springer-Verlag, New York
			* Walter Rudin. ''Real and Complex Analysis'' (1987), 3rd Ed., McGraw Hill, New York.
			* [[Elias Stein]], ''Singular integrals and differentiability properties of functions'' (1970), Princeton University Press, Princeton.

			[[Category:Theorems in complex analysis]]

en>Mcld: /* Efficient computation of reassigned times and frequencies */ deduplicate and clarify a para

2012-08-10T15:12:43Z

Efficient computation of reassigned times and frequencies: deduplicate and clarify a para

New page

Nice to satisfy you, my title is Ling and I completely dig that title. Some time in the past he selected to live in Idaho. The thing she adores most is to perform handball but she can't make it her profession. The job he's been occupying for years is a messenger.<br><br>Visit my web blog :: extended car warranty ([http://Sticker-Shocked.com/UserProfile/tabid/43/userId/404/Default.aspx linked web page])