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Direct Theorem about Approximation on a Family of Two Segments

Authors:	K G Mezhevich N A Shirokov

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.

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