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Abstract  

The trigonometric polynomials of Fejér and Young are defined by

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}}$$ \end{document}
and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}$$ \end{document}
, respectively. We prove that the inequality
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}}$$ \end{document}
holds for all n ≥ 2 and x ∈ (0, π). The lower bound is sharp.

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LetT n (x) be trigonometric polynomial of degreen, and be conjugation ofT n (x). In this paper we obtain the complete asymptotic expansion for

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_n = \mathop {\sup }\limits_{\left\| {T_n } \right\|C \leqslant 1} \left\| {\tilde T_n } \right\|_C$$ \end{document}
forn→∞.

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Abstract  

Let σ > 0. For 1 ≦ p ≦ ∞, the Bernstein space B σ p is a Banach space of all fL p (ℝ) such that f is bandlimited to σ; that is, the distributional Fourier transform of f is supported in [−σ,σ]. We study the approximation of fB σ p by finite trigonometric sums

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_\tau (x) = \chi _\tau (x) \cdot \sum\limits_{|k| \leqq \sigma \tau /\pi } {c_{k,\tau } e^{i\frac{\pi } {\tau }kx} }$$ \end{document}
in L p norm on ℝ as τ → ∞, where χ τ denotes the indicator function of [−τ, τ].

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В статье получены оце нки скорости приближ ений в равномерной метрике классов функций с ограниченной смеша нной производной тригонометрическим и полиномами с заданн ым числом гармоник. Даны приложения к оценкамε-этропии этих классов. Из получ енных результатов вытекают в некоторых случаях известные ре зультаты.

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