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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1998, Volume 64, Issue 2, Pages 251–259 (Mi mz1393)

Some properties of rational approximations of degree $(k,1)$ in the Hardy space $H_2(\mathscr D)$

M. A. Nazarenko

M. V. Lomonosov Moscow State University

Abstract: We prove that the well-known interpolation conditions for rational approximations with free poles are not sufficient for finding a rational function of the least deviation. For rational approximations of degree $(k,1)$, we establish that these interpolation conditions are equivalent to the assertion that the interpolation point $c$ is a stationary point of the function $\Omega_k(c)$ defined as the squared deviation of $f$ from the subspace of rational functions with numerator of degree $\leq k$ and with a given pole $1/\overline c$. For any positive integers $k$ and $s$, we construct a function $g\in H_2(\mathscr D)$ such that $R_{k,1}(g)=R_{k+s,1}(g)>0$. where $R_{k,1}(g)$ is the least deviation of $g$ from the class of rational function of degree $\leq (k,1)$.

DOI: https://doi.org/10.4213/mzm1393

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English version:
Mathematical Notes, 1998, 64:2, 213–219

Bibliographic databases:

UDC: 517.538.5
Revised: 26.05.1997

Citation: M. A. Nazarenko, “Some properties of rational approximations of degree $(k,1)$ in the Hardy space $H_2(\mathscr D)$”, Mat. Zametki, 64:2 (1998), 251–259; Math. Notes, 64:2 (1998), 213–219

Citation in format AMSBIB
\Bibitem{Naz98} \by M.~A.~Nazarenko \paper Some properties of rational approximations of degree $(k,1)$ in the Hardy space $H_2(\mathscr D)$ \jour Mat. Zametki \yr 1998 \vol 64 \issue 2 \pages 251--259 \mathnet{http://mi.mathnet.ru/mz1393} \crossref{https://doi.org/10.4213/mzm1393} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1680937} \zmath{https://zbmath.org/?q=an:0926.41010} \transl \jour Math. Notes \yr 1998 \vol 64 \issue 2 \pages 213--219 \crossref{https://doi.org/10.1007/BF02310308} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000078147600029}