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 contains the following theorem presented also in Joseph L. Walsh's well-known monograph (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


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

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 $$\left.\right. A^2_0$$ became common for the class $$\left.\right. H_2$$.

In 1948 Mkhitar Djrbashian 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


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

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.