Geodesic
Some properties of rational approximations of degree $(k,1)$ in the Hardy space $H_2(\mathscr D)$
Matematičeskie zametki, Tome 64 (1998) no. 2, pp. 251-259 
M. A. Nazarenko. Some properties of rational approximations of degree $(k,1)$ in the Hardy space $H_2(\mathscr D)$. Matematičeskie zametki, Tome 64 (1998) no. 2, pp. 251-259. http://geodesic.mathdoc.fr/item/MZM_1998_64_2_a11/
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     title = {Some properties of rational approximations of degree $(k,1)$ in the {Hardy} space $H_2(\mathscr D)$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\leq (k,1)$.

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