Direct Theorem about Approximation on a Family of Two Segments

Abstract

Introduce the notation: $E_\varepsilon ,0 < \varepsilon \leqslant 1$, is the union of two segments [-1,1] and [-1 +$i\varepsilon$,1+$i\varepsilon$], $\alpha$ is a noninteger number,$\Lambda _\alpha (E_\varepsilon )$ is the Hölder class with exponent $\alpha$ on $E_\varepsilon$ The following result announced by the authors in [J. Math. Sci. 117 (2003), No. 3] is proved. There exist numbers a ₁ ($\alpha$) , b ₁ ($\alpha$) $>$ 0 depending only on $\alpha$ such that for any $0 < \varepsilon \leqslant 1,f \in \Lambda ^\alpha (E_\varepsilon ),||f||_{\Lambda ^\alpha (E_\varepsilon )} \leqslant 1$ there exists a polynomial$P_n (z,\varepsilon ),{\text{ deg }}P_n \leqslant n$, such that $|f(z) - P_n (z,\varepsilon )| \leqslant a_1 (\alpha ) \cdot e^{\frac{{b_1 (\alpha )}}{\varepsilon }} \cdot \rho _{1/n}^\alpha (z,\varepsilon )$. Bibliography: 11 titles.