Wirtinger's representation and projection theorem

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 subspace $\left.\right.H_{2}$ 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<+\infty ,

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)={\frac {1}{\pi }}\iint _{|\zeta |<1}F(\zeta ){\frac {dS}{(1-{\overline {\zeta }}z)^{2}}},\quad |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 $\left.\right.A_{0}^{2}$ 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 $\left.\right.A_{\alpha }^{2}$ of functions $\left.\right.f(z)$ holomorphic in $\left.\right.|z|<1$ , which satisfy the condition

\|f\|_{A_{\alpha }^{2}}=\left\{{\frac {1}{\pi }}\iint _{|z|<1}|f(z)|^{2}(1-|z|^{2})^{\alpha -1}\,dS\right\}^{1/2}<+\infty {\text{ for some }}\alpha \in (0,+\infty ),

and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted $\left.\right.A_{\omega }^{2}$ 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]

References

^ Wirtinger, W. (1932). "Uber eine Minimumaufgabe im Gebiet der analytischen Functionen". Monatshefte für Mathematik und Physik. 39: 377–384. doi:10.1007/bf01699078. S2CID 120529823.
^ Walsh, J. L. (1956). "Interpolation and Approximation by Rational Functions in the Complex Domain". Amer. Math. Soc. Coll. Publ. XX. Ann Arbor, Michigan: Edwards Brothers, Inc.
^ Djrbashian, M. M. (1948). "On the Representability Problem of Analytic Functions" (PDF). Soobsch. Inst. Matem. I Mekh. Akad. Nauk Arm. SSR. 2: 3–40.
^ Jerbashian, A. M. (2005). "On the Theory of Weighted Classes of Area Integrable Regular Functions". Complex Variables. 50 (3): 155–183. doi:10.1080/02781070500032846. S2CID 218556016.

[1] Wirtinger, W. (1932). "Uber eine Minimumaufgabe im Gebiet der analytischen Functionen". Monatshefte für Mathematik und Physik. 39: 377–384. doi:10.1007/bf01699078. S2CID 120529823.

[2] Walsh, J. L. (1956). "Interpolation and Approximation by Rational Functions in the Complex Domain". Amer. Math. Soc. Coll. Publ. XX. Ann Arbor, Michigan: Edwards Brothers, Inc.

[3] Djrbashian, M. M. (1948). "On the Representability Problem of Analytic Functions" (PDF). Soobsch. Inst. Matem. I Mekh. Akad. Nauk Arm. SSR. 2: 3–40.

[4] Jerbashian, A. M. (2005). "On the Theory of Weighted Classes of Area Integrable Regular Functions". Complex Variables. 50 (3): 155–183. doi:10.1080/02781070500032846. S2CID 218556016.

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See also

References