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

Content Provider	Springer Nature Link
Author	Ramazav, A. K.
Copyright Year	2000
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).
Starting Page	613
Ending Page	627
Page Count	15
File Format	PDF
ISSN	00014346
Journal	Mathematical Notes
Volume Number	66
Issue Number	5
e-ISSN	15738876
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	2000-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	polyanalytic function direct sum of orthogonal subspaces rational approximation extremum problem Bessel’s inequality polyrational function Mathematics
Content Type	Text
Resource Type	Article

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