en>Lemnaminor: Disambiguated: Growth curve → Growth curve (statistics)

2014-12-02T09:22:57Z

Disambiguated: Growth curve → Growth curve (statistics)

en>David Eppstein: /* Example uses */ Supply requested citations

2014-01-30T23:04:58Z

Example uses: Supply requested citations

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	~~Greetings! I am Myrtle Shroyer~~. ~~Hiring~~ is ~~her day occupation now and she will~~ not ~~alter it anytime quickly~~. ~~For many years he~~'~~s been residing~~ in ~~North Dakota~~ and ~~his family members loves it~~. ~~One of~~ the ~~extremely best issues in~~ the ~~world~~ for me is to ~~do aerobics and I~~'~~ve been performing it~~ for ~~fairly a while~~.<br><br>~~My website~~ [~~http:~~//~~rtdcs~~.~~hufs~~.ac.~~kr/?document_srl=780411 at home std testing~~]		{{refimprove\|date=October 2010}}
			In [[mathematics]], a '''dual wavelet''' is the [[dual space\|dual]] to a [[wavelet]]. In general, the [[wavelet series]] generated by a [[square integrable]] [[function (mathematics)\|function]] will have a dual series, in the sense of the [[Riesz representation theorem]]. However, the dual series is not in general representable by a square integral function itself.

			==Definition==
			Given a square integrable function <math>\psi\in L^2(\mathbb{R})</math>, define the series <math>\{\psi_{jk}\}</math> by

			:<math>\psi_{jk}(x) = 2^{j/2}\psi(2^jx-k)</math>

			for integers <math>j,k\in \mathbb{Z}</math>.

			Such a function is called an '''''R''-function''' if the linear span of <math>\{\psi_{jk}\}</math> is [[dense set\|dense]] in <math>L^2(\mathbb{R})</math>, and if there exist positive constants ''A'', ''B'' with <math>0<A\leq B < \infty</math> such that

			:<math>A \Vert c_{jk} \Vert^2_{l^2} \leq
			\bigg\Vert \sum_{jk=-\infty}^\infty c_{jk}\psi_{jk}\bigg\Vert^2_{L^2} \leq
			B \Vert c_{jk} \Vert^2_{l^2}\,</math>

			for all bi-infinite [[square summable]] series <math>\{c_{jk}\}</math>. Here, <math>\Vert \cdot \Vert_{l^2}</math> denotes the square-sum norm:

			:<math>\Vert c_{jk} \Vert^2_{l^2} = \sum_{jk=-\infty}^\infty \vert c_{jk}\vert^2</math>

			and <math>\Vert \cdot\Vert_{L^2}</math> denotes the usual norm on <math>L^2(\mathbb{R})</math>:

			:<math>\Vert f\Vert^2_{L^2}= \int_{-\infty}^\infty \vert f(x)\vert^2 dx</math>

			By the [[Riesz representation theorem]], there exists a unique dual basis <math>\psi^{jk}</math> such that

			:<math>\langle \psi^{jk} \vert \psi_{lm} \rangle = \delta_{jl} \delta_{km}</math>

			where <math>\delta_{jk}</math> is the [[Kronecker delta]] and <math>\langle f\vert g \rangle</math> is the usual [[inner product]] on <math>L^2(\mathbb{R})</math>. Indeed, there exists a unique [[series representation]] for a square integrable function ''f'' expressed in this basis:

			:<math>f(x) = \sum_{jk} \langle \psi^{jk} \vert f \rangle \psi_{jk}(x)</math>

			If there exists a function <math>\tilde{\psi} \in L^2(\mathbb{R})</math> such that

			:<math>\tilde{\psi}_{jk} = \psi^{jk}</math>

			then <math>\tilde{\psi}</math> is called the '''dual wavelet''' or the '''wavelet dual to ψ'''. In general, for some given ''R''-function ψ, the dual will not exist. In the special case of <math>\psi = \tilde{\psi}</math>, the wavelet is said to be an [[orthogonal wavelet]].

			An example of an ''R''-function without a dual is easy to construct. Let <math>\phi</math> be an orthogonal wavelet. Then define <math>\psi(x) = \phi(x) + z\phi(2x)</math> for some complex number ''z''. It is straightforward to show that this ψ does not have a wavelet dual.

			==See also==
			* [[Multiresolution analysis]]

			==References==
			* Charles K. Chui, ''An Introduction to Wavelets (Wavelet Analysis & Its Applications)'', (1992), Academic Press, San Diego, ISBN 0-12-174584-8

			[[Category:Wavelets]]
			[[Category:Duality theories\|Wavelet]]

en>ZéroBot: Robot: Adding ru:Функция Гомпертца

2012-07-18T16:49:27Z

Robot: Adding ru:Функция Гомпертца

New page

Greetings! I am Myrtle Shroyer. Hiring is her day occupation now and she will not alter it anytime quickly. For many years he's been residing in North Dakota and his family members loves it. One of the extremely best issues in the world for me is to do aerobics and I've been performing it for fairly a while.<br><br>My website [http://rtdcs.hufs.ac.kr/?document_srl=780411 at home std testing]

Gompertz function - Revision history

en>Lemnaminor: Disambiguated: Growth curve → Growth curve (statistics)

en>David Eppstein: /* Example uses */ Supply requested citations

en>ZéroBot: Robot: Adding ru:Функция Гомпертца