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 {\displaystyle \left.\right.H_{2}} of the simple, unweighted holomorphic Hilbert space L 2 {\displaystyle \left.\right.L^{2}} of functions square-integrable over the surface of the unit disc { z : | z | < 1 } {\displaystyle \left.\right.\{z:|z|<1\}} of the complex plane, along with a form of the orthogonal projection from L 2 {\displaystyle \left.\right.L^{2}} to H 2 {\displaystyle \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 F ( z ) {\displaystyle \left.\right.\left.F(z)\right.} is of the class L 2 {\displaystyle \left.\right.L^{2}} on | z | < 1 {\displaystyle \left.\right.|z|<1} , i.e.
∬ | z | < 1 | F ( z ) | 2 d S < + ∞ , {\displaystyle \iint _{|z|<1}|F(z)|^{2}\,dS<+\infty ,}
where d S {\displaystyle \left.\right.dS} is the area element, then the unique function f ( z ) {\displaystyle \left.\right.f(z)} of the holomorphic subclass H 2 ⊂ L 2 {\displaystyle H_{2}\subset L^{2}} , such that
∬ | z | < 1 | F ( z ) − f ( z ) | 2 d S {\displaystyle \iint _{|z|<1}|F(z)-f(z)|^{2}\,dS}
is least, is given by
f ( z ) = 1 π ∬ | ζ | < 1 F ( ζ ) d S ( 1 − ζ ¯ z ) 2 , | z | < 1. {\displaystyle f(z)={\frac {1}{\pi }}\iint _{|\zeta |<1}F(\zeta ){\frac {dS}{(1-{\overline {\zeta }}z)^{2}}},\quad |z|<1.}
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