Wirtinger's representation and projection theorem explained

\left.\right. H _{2} of the simple, unweighted holomorphic Hilbert space \left.\right. L ^{2} of functions square-integrable over the surface of the unit disc \left.\right.\{z:|z|< 1 \} of the complex plane, along with a form of the orthogonal projection from \left.\right. L ^{2} to \left.\right. H _{2} .

Wirtinger's paper ^[1] contains the following theorem presented also in Joseph L. Walsh's well-known monograph^[2] (p. 150) with a different proof. If

\left.\right.\left. F(z)\right. is of the class \left.\right. L ^{2} on \left.\right. |z|< 1 , i.e.\iint _{|z|<1} |F(z)| ^{2} dS< +inf ty,

where

\left.\right. dS is the area element, then the unique function \left.\right. f(z) of the holomorphic subclass H _{2\subset} L ^{2} , such that\iint _{|z|<1} |F(z) - f(z)| ^{2} dS

is least, is given by

f(z) = (1 \pi\iint)/(F(\zeta) _{|\zeta|<1}) (dS)/(( 1- \overline\zeta z) ^{2}) , |z|< 1 .

The last formula gives a form for the orthogonal projection from

\left.\right. L ^{2} to \left.\right. H _{2} . Besides, replacement of \left.\right. F(\zeta) by \left.\right. f(\zeta) makes it Wirtinger's representation for all f(z)\in H _{2} . 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 \left.\right. A ^{2} _{0} became common for the class \left.\right. H _{2} .

In 1948 Mkhitar Djrbashian^[3] extended Wirtinger's representation and projection to the wider, weighted Hilbert spaces

\left.\right. A ^{2} _{\alpha} of functions \left.\right. f(z) holomorphic in \left.\right.|z|< 1 , which satisfy the condition\|f\| _{A ^{2} _{\alpha}} = \left\{ (1 \pi\iint)/(_{|z|<1}) |f(z)| ^{2(1-|z|} ^{2)} ^{\alpha-1} dS\right\} ^{1/2} < +inf ty forsome \alpha\in( 0 , +inf ty),

and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted

\left.\right. A ^{2} _{\omega} spaces of functions holomorphic in \left.\right. |z|< 1 and similar spaces of entire functions, the unions of which respectively coincide with all functions holomorphic in \left.\right. |z|< 1 and the whole set of entire functions can be seen in.^[4]

See also

• A. M.. Jerbashian. V. S. Zakaryan . The Contemporary Development in M. M. Djrbashian Factorization Theory and Related Problems of Analysis. Izv. NAN of Armenia, Matematika (English translation: Journal of Contemporary Mathematical Analysis). 44. 6. 2009.

Notes and References

1. W. . Wirtinger. Uber eine Minimumaufgabe im Gebiet der analytischen Functionen. Monatshefte für Mathematik und Physik. 39. 377–384. 1932 . 10.1007/bf01699078. 120529823.
2. J. L.. Walsh. Interpolation and Approximation by Rational Functions in the Complex Domain. Amer. Math. Soc. Coll. Publ. XX. Edwards Brothers, Inc.. Ann Arbor, Michigan. 1956.
3. M. M.. Djrbashian. Mkhitar Djrbashian. On the Representability Problem of Analytic Functions. Soobsch. Inst. Matem. I Mekh. Akad. Nauk Arm. SSR. 2. 3–40. 1948.
4. A. M.. Jerbashian. On the Theory of Weighted Classes of Area Integrable Regular Functions. Complex Variables. 50. 155–183. 2005. 3. 10.1080/02781070500032846. 218556016.