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 [math]\displaystyle{ \left.\right. H_2 }[/math] of the simple, unweighted holomorphic Hilbert space [math]\displaystyle{ \left.\right. L^2 }[/math] of functions square-integrable over the surface of the unit disc [math]\displaystyle{ \left.\right.\{z:|z|\lt 1\} }[/math] of the complex plane, along with a form of the orthogonal projection from [math]\displaystyle{ \left.\right. L^2 }[/math] to [math]\displaystyle{ \left.\right. H_2 }[/math].

Wirtinger's paper contains the following theorem presented also in Joseph L. Walsh's well-known monograph (p. 150) with a different proof. If [math]\displaystyle{ \left.\right.\left. F(z)\right. }[/math] is of the class [math]\displaystyle{ \left.\right. L^2 }[/math] on [math]\displaystyle{ \left.\right. |z|\lt 1 }[/math], i.e.

[math]\displaystyle{ \iint_{|z|\lt 1}|F(z)|^2 \, dS\lt +\infty, }[/math]

where [math]\displaystyle{ \left.\right. dS }[/math] is the area element, then the unique function [math]\displaystyle{ \left.\right. f(z) }[/math] of the holomorphic subclass [math]\displaystyle{ H_2\subset L^2 }[/math], such that

[math]\displaystyle{ \iint_{|z|\lt 1}|F(z)-f(z)|^2 \, dS }[/math]

is least, is given by

[math]\displaystyle{ f(z)=\frac1\pi\iint_{|\zeta|\lt 1}F(\zeta)\frac{dS}{(1-\overline\zeta z)^2},\quad |z|\lt 1. }[/math]

The last formula gives a form for the orthogonal projection from [math]\displaystyle{ \left.\right. L^2 }[/math] to [math]\displaystyle{ \left.\right. H_2 }[/math]. Besides, replacement of [math]\displaystyle{ \left.\right. F(\zeta) }[/math] by [math]\displaystyle{ \left.\right. f(\zeta) }[/math] makes it Wirtinger's representation for all [math]\displaystyle{ f(z)\in H_2 }[/math]. 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 [math]\displaystyle{ \left.\right. A^2_0 }[/math] became common for the class [math]\displaystyle{ \left.\right. H_2 }[/math].

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

[math]\displaystyle{ \|f\|_{A^2_\alpha}=\left\{\frac1\pi\iint_{|z|\lt 1}|f(z)|^2(1-|z|^2)^{\alpha-1} \, dS\right\}^{1/2}\lt +\infty\text{ for some }\alpha\in(0,+\infty), }[/math]

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

See also

• Jerbashian, A. M.; V. S. Zakaryan (2009). "The Contemporary Development in M. M. Djrbashian Factorization Theory and Related Problems of Analysis". Izv. NAN of Armenia, Matematika (English translation: Journal of Contemporary Mathematical Analysis) 44 (6).