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.\$ 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.

: $\iint_, F(z), ^2 \, dS<+\infty,$

''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''

: $\iint_, F(z)-f(z), ^2 \, dS$

''is least, is given by

: $f(z)=\frac1\pi\iint_F(\zeta)\frac,\quad , z, <1.$

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

: $\, f\, _=\left\^<+\infty\text\alpha\in(0,+\infty),$

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.

See also

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References

{{Functional analysis

Theorems in complex analysis
Theorems in functional analysis
Theorems in approximation theory