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

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 ${\displaystyle \left.\right.\left.F(z)\right.}$ is of the class ${\displaystyle \left.\right.L^{2}}$ on ${\displaystyle \left.\right.|z|<1}$, i.e.

${\displaystyle \iint _{|z|<1}|F(z)|^{2}\,dS<+\infty ,}$

where ${\displaystyle \left.\right.dS}$ is the area element, then the unique function ${\displaystyle \left.\right.f(z)}$ of the holomorphic subclass ${\displaystyle H_{2}\subset L^{2}}$, such that

${\displaystyle \iint _{|z|<1}|F(z)-f(z)|^{2}\,dS}$

is least, is given by

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

The last formula gives a form for the orthogonal projection from ${\displaystyle \left.\right.L^{2}}$ to ${\displaystyle \left.\right.H_{2}}$. Besides, replacement of ${\displaystyle \left.\right.F(\zeta )}$ by ${\displaystyle \left.\right.f(\zeta )}$ makes it Wirtinger's representation for all ${\displaystyle f(z)\in H_{2}}$. 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 ${\displaystyle \left.\right.A_{0}^{2}}$ became common for the class ${\displaystyle \left.\right.H_{2}}$.

In 1948 Mkhitar Djrbashian[3] extended Wirtinger's representation and projection to the wider, weighted Hilbert spaces ${\displaystyle \left.\right.A_{\alpha }^{2}}$ of functions ${\displaystyle \left.\right.f(z)}$ holomorphic in ${\displaystyle \left.\right.|z|<1}$, which satisfy the condition

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

and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted ${\displaystyle \left.\right.A_{\omega }^{2}}$ spaces of functions holomorphic in ${\displaystyle \left.\right.|z|<1}$ and similar spaces of entire functions, the unions of which respectively coincide with all functions holomorphic in ${\displaystyle \left.\right.|z|<1}$ and the whole set of entire functions can be seen in.[4]