Geodesic
Schur’s rational approximation of Schur’s functions
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 4 (2011), pp. 63-72
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The problem of approximating elements from the class $H_2^+$ of the analytic functions in the closed unit disk $U$ assuming only real values on the segment [0,1] is investigated. As approximant class is taken to be ${\mathcal H}_{n}^{+}$ which is the class of irreducible real rational functions with the degrees of a numerator and a denominator not greater $n$. It is proved that if $f\in H_2^+$ and $f\notin {\mathcal H}_{k}^{+}$ where $k then any local minimizer of nonlinear programme $\displaystyle \|{f-g}\|^2\longrightarrow \min_{g\in {\mathcal H}_{n}^{+}}$ does not belong to ${\mathcal H}_{m}^{+}$, where $m. The result is expanded to the class $S^+$ of Schur's functions selected from $H_2^+$ by the condition $\sup_{z\in U} |f(z)|\leq 1$. If ${\mathcal S}_n^+$ is a Schur's subclass of ${\mathcal H}_{n}^{+}$ then it is proved that, when $f\in S^+$ and $f\notin {\mathcal S}_{k}^{+}$, where $k, any local minimizer of non linear programme $\displaystyle \|{f-g}\|^2\longrightarrow \min_{g\in {\mathcal S}_{n}^{+}}$ does not belong to ${\mathcal S}_{m}^{+}$, where $m.
Keywords: unit disk, Schur’s function, approximation, rational function, Schur’s algorithm. 
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     title = {Schur{\textquoteright}s rational approximation of {Schur{\textquoteright}s} functions},
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V. S. Mikheev. Schur’s rational approximation of Schur’s functions. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 4 (2011), pp. 63-72. http://geodesic.mathdoc.fr/item/VSPUI_2011_4_a6/

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