Wirtinger's representation and projection theorem
# 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

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

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 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.

