Wirtinger's representation and projection theorem

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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 of the simple, unweighted holomorphic Hilbert space of functions square-integrable over the surface of the unit disc of the complex plane, along with a form of the orthogonal projection from to .

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

where is the area element, then the unique function of the holomorphic subclass , such that

is least, is given by

The last formula gives a form for the orthogonal projection from to . Besides, replacement of by makes it Wirtinger's representation for all . 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 became common for the class .

In 1948 Mkhitar Djrbashian[3] extended Wirtinger's representation and projection to the wider, weighted Hilbert spaces of functions holomorphic in , which satisfy the condition

and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted spaces of functions holomorphic in and similar spaces of entire functions, the unions of which respectively coincide with all functions holomorphic in and the whole set of entire functions can be seen in.[4]

See also[edit]

  • 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). 

References[edit]

  1. ^ Wirtinger, W. (1932). "Uber eine Minimumaufgabe im Gebiet der analytischen Functionen". Monatshefte fur Math. und Phys. 39: 377–384. doi:10.1007/bf01699078. 
  2. ^ Walsh, J. L. (1956). "Interpolation and Approximation by Rational Functions in the Complex Domain". Amer. Math. Soc. Coll. Publ. XX. Ann Arbor, Michigan: Edwards Brothers, Inc. 
  3. ^ Djrbashian, M. M. (1948). "On the Representability Problem of Analytic Functions" (PDF). Soobsch. Inst. Matem. i Mekh. Akad. Nauk Arm. SSR. 2: 3–40. 
  4. ^ Jerbashian, A. M. (2005). "On the Theory of Weighted Classes of Area Integrable Regular Functions". Complex Variables. 50: 155–183. doi:10.1080/02781070500032846.