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 $H_2$ of the simple, unweighted holomorphic Hilbert space $L^2$ of functions square-integrable over the surface of the unit disc $\{z:|z|<1\}$ of the complex plane, along with a form of the orthogonal projection from $L^2$ to $H_2$.

Wirtinger's paper contains the following theorem presented also in Joseph L. Walsh's well-known monograph (p. 150) with a different proof. If $F(z)$ is of the class $L^2$ on $|z|<1$, i.e.

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

where $dS$ is the area element, then the unique function $f(z)$ of the holomorphic subclass $H_2\subset L^2$, such that

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

is least, is given by

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

The last formula gives a form for the orthogonal projection from $L^2$ to $H_2$. Besides, replacement of $F(\zeta )$ by $f(\zeta )$ 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 $A_0^2$ became common for the class $H_2$.

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

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

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

See also

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