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.
∬ | z | < 1 | F ( z ) | 2 d S < + ∞ , where d S is the area element, then the unique function f ( z ) of the holomorphic subclass H 2 ⊂ L 2 , such that
∬ | z | < 1 | F ( z ) − f ( z ) | 2 d S is least, is given by
f ( z ) = 1 π ∬ | ζ | < 1 F ( ζ ) d S ( 1 − ζ ¯ z ) 2 , | z | < 1. The last formula gives a form for the orthogonal projection from L 2 to H 2 . Besides, replacement of F ( ζ ) by f ( ζ ) makes it Wirtinger's representation for all f ( z ) ∈ 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 α 2 of functions f ( z ) holomorphic in | z | < 1 , which satisfy the condition
∥ f ∥ A α 2 = { 1 π ∬ | z | < 1 | f ( z ) | 2 ( 1 − | z | 2 ) α − 1 d S } 1 / 2 < + ∞ for some α ∈ ( 0 , + ∞ ) , and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted A ω 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.
