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Representation of the space of polyanalytic functions as a direct sum of orthogonal subspaces. Application to rational approximations

Authors:

A K Ramazanov

Institution:

(1) Kaluga Branch of N. é. Bauman Moscow State Technical University, Moscow, USSR

Abstract:

Suppose thatD={z:|z|<1}, L ₂ (D) is the space of functions square-integrable over area inD,A _k (D) is the set of allk-analytic functions inD, (A ₁ (D)=A(D) is the set of all analytic functions inD),A _k L ₂ (D)=L ₂ (D)∩A _k (D),A ₁ L ₂ (D)=AL ₂ (D),

$A_k L_2^0 \left( D \right) = \left\{ {f {\text{:}} f(z) = \frac{{\partial ^{k - 1} }}{{\partial z^{k - 1} }}\left( {\left( {1 - z\bar z} \right)^{k - 1} F\left( z \right)} \right); F \in A\left( D \right), f \in A_k L_2 \left( D \right)} \right\}$

. It is proved that the subspacesA _k L ₂ ⁰ (D),k=1, 2,..., are orthogonal to one another and the spaceA _m L ₂ (D) is the direct sum of such subspaces fork=1, 2,...,m. The kernel of the orthogonal projection operator from the spaceA _m L ₂ (D) onto its subspacesA _k L ₂ ⁰ (D) is obtained. These results are applied to the study of the properties of polyrational functions of best approximation in the metricL ₂ (D). Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 741–759, November, 1999.

Keywords:

polyanalytic function direct sum of orthogonal subspaces rational approximation extremum problem Bessel’ s inequality polyrational function

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