en>SoledadKabocha: Disambiguation link repair: hydron → hydron (chemistry)

2013-06-22T04:56:06Z

Disambiguation link repair: hydron → hydron (chemistry)

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{{multiple issues|
{{Expert-subject|Mathematics|date=May 2010}}
{{Technical|date=May 2010}}
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In mathematics, '''Wirtinger's representation and projection theorem''' is a [[theorem]] proved by [[Wilhelm Wirtinger]] in 1932 in connection with some problems of [[approximation theory]]. This theorem gives the representation formula for the [[holomorphic]] [[Linear subspace|subspace]] <math>\left.\right. H_2 </math> of the simple, unweighted holomorphic [[Hilbert space]] <math>\left.\right. L^2 </math> of functions [[square-integrable]] over the surface of the unit disc <math>\left.\right.\{z:|z|<1\} </math> of the [[complex plane]], along with a form of the [[orthogonal projection]] from <math>\left.\right. L^2 </math> to <math>\left.\right. H_2 </math>.

Wirtinger's paper <ref>{{Cite journal
|first=W. |last= Wirtinger
|title=Uber eine Minimumaufgabe im Gebiet der analytischen Functionen
|journal=Monatshefte fur Math. und Phys.,
|volume=39
|pages=377–384
|year=1932}}</ref> contains the following theorem presented also in [[Joseph L. Walsh]]'s well-known monograph
<ref>{{Cite journal
|first=J. L.|last= Walsh
|title=Interpolation and Approximation by Rational Functions in the Complex Domain
|journal=Amer. Math. Soc. Coll. Publ. XX
|publisher=Edwards Brothers, Inc.
|location=Ann Arbor, Michigan
|year=1956}}</ref>
(p. 150) with a different proof. ''If'' <math>\left.\right.\left. F(z)\right.</math> ''is of the class'' <math>\left.\right. L^2 </math> on <math>\left.\right. |z|<1 </math>, ''i.e.

: <math> \iint_{|z|<1}|F(z)|^2 \, dS<+\infty,</math>

''where <math>\left.\right. dS </math> is the [[area element]], then the unique function <math>\left.\right. f(z)</math> of the holomorphic subclass <math> H_2\subset L^2 </math>, such that''

: <math> \iint_{|z|<1}|F(z)-f(z)|^2 \, dS </math>

''is least, is given by

: <math> f(z)=\frac1\pi\iint_{|\zeta|<1}F(\zeta)\frac{dS}{(1-\overline\zeta z)^2},\quad |z|<1. </math>

The last formula gives a form for the orthogonal projection from <math>\left.\right. L^2 </math> to <math>\left.\right. H_2 </math>. Besides, replacement of <math> \left.\right. F(\zeta) </math> by <math>\left.\right. f(\zeta) </math> makes it Wirtinger's representation for all <math>f(z)\in H_2 </math>. This is an analog of the well-known [[Cauchy integral formula]] with the square of the Cauchy kernel. Later, after the 1950s, a degree of the Cauchy kernel was called [[reproducing kernel]], and the notation <math>\left.\right. A^2_0</math> became common for the class <math>\left.\right. H_2</math>.

In 1948 [[Mkhitar Djrbashian]]<ref>{{Cite journal
|first=M. M.|last=Djrbashian|authorlink=Mkhitar Djrbashian
|title=On the Representability Problem of Analytic Functions
|journal=Soobsch. Inst. Matem. i Mekh. Akad. Nauk Arm. SSR
|volume=2
|pages=3–40
|year=1948
|url=http://math.sci.am/upload/file/ArmenJerbashian/1945-1948.pdf}}</ref> extended Wirtinger's representation and projection to the wider, weighted Hilbert spaces <math>\left.\right. A^2_\alpha </math> of functions <math>\left.\right. f(z)</math> holomorphic in <math> \left.\right.|z|<1</math>, which satisfy the condition

: <math>\|f\|_{A^2_\alpha}=\left\{\frac1\pi\iint_{|z|<1}|f(z)|^2(1-|z|^2)^{\alpha-1} \, dS\right\}^{1/2}<+\infty\text{ for some }\alpha\in(0,+\infty),</math>

and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted <math>\left.\right. A^2_\omega</math> spaces of functions holomorphic in <math>\left.\right. |z|<1</math> and similar spaces of entire functions, the unions of which respectively coincide with ''all'' functions holomorphic in <math>\left.\right. |z|<1</math> and the ''whole'' set of entire functions can be seen in.<ref>{{Cite journal
|first=A. M.|last=Jerbashian
|title=On the Theory of Weighted Classes of Area Integrable Regular Functions
|journal=Complex Variables
|volume=50
|pages=155–183
|year=2005}}</ref>

==See also==
* {{Cite journal
|first=A. M.|last=Jerbashian
|coauthor=V. S. Zakaryan
|title=The Contemporary Development in M. M. Djrbashian Factorization Theory and Related Problems of Analysis
|journal=Izv. NAN of Armenia, Matematika (English translation: Journal of Contemporary Mathematical Analysis)
|volume=44|number=6|year=2009}}

==References==
{{Reflist}}

[[Category:Theorems in complex analysis]]
[[Category:Theorems in functional analysis]]
[[Category:Theorems in approximation theory]]

Bjerrum plot - Revision history

en>SoledadKabocha: Disambiguation link repair: hydron → hydron (chemistry)