Polynomial Approximations on a Family of Two Segments

Abstract

Introduce the notation:$E_\varepsilon ,{\text{ }}0 < \varepsilon < 1,$,is the union of two segments [- 1,1] and $\left[ { - 1 + i\varepsilon ,1 + i\varepsilon } \right],\Lambda _\alpha (E_\varepsilon )$ is the Holder class with exponent α on $E_\varepsilon ,{\text{ }}0 < \alpha < 1,{\text{ }}g_\varepsilon (z)$ is the Green function of the set $C\backslash E_\varepsilon$ with a logarithmic pole at infinity, $L_h (\varepsilon ) = \{ z \in C\backslash E_\varepsilon :g_\varepsilon (z) = h\} ,h >0,\rho _h (z,\varepsilon ) = {\text{dist(}}z,L_h (\varepsilon ))$. We prove the following result: There exist positive constants b(α)and a(α) depending only on α such that if

$$\left| {f(z) - P_n (z)} \right| \leqslant c(\alpha ,\varepsilon ) \cdot \rho _{\frac{1}{h}}^\alpha (z,\varepsilon ),{\text{ }}z \in E_\varepsilon .$$

then $c(\alpha ,\varepsilon ) \geqslant a(\alpha ) \cdot e^{\frac{{b(\alpha )}}{\varepsilon }} ,0 < \varepsilon \leqslant 1$.Bibliography: 3 titles.