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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
Received: 30.01.1995
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
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\jour Math. Notes
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\pages 213--219
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