Schur’s rational approximation of Schur’s functions

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 $k<n$ 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 $m<n$. 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 $k<n$, 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 $m<n$.