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  • Author or Editor: S. P. Zhou x
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Abstract  

We generalize some old and new results on the determination of jumps of a periodic, Lebesgue integrable function f at each point of discontinuity of first kind in terms of the partial sums of the conjugate series to the Fourier series of f in [1], and in terms of the Abel-Poisson means in [2], to some more general linear operators which satisfy some certain conditions. The linear operators in discussion include the Fejér means, de la Vallée-Poussin means, and Bernstein-Rogosinski sums.

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The present note will present a different and direct way to generalize the convexity while keep the classical results for L 1-convergence still alive.

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Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min [5] to include L p spaces for 1≤ p ≤ ∞ while loosing the restriction ρ > 2 at the same time.

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Summary  

To verify the universal validity of the ``two-sided'' monotonicity condition introduced in [4], we will apply it to include more classical examples. The present paper selects the L p convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.

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Abstract

By employing new ideas and techniques, we will refigure out the whole frame of L 1-approximation. First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement fL 2π but directly consider the sine or cosine series. Secondly, to achieve nontrivial generalizations in complex spaces, we use a one-sided condition with some kind of balance conditions. In addition, a conjecture raised in [9] is disproved in Section 3.

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Abstract  

Let
\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} $$a = e^{ - 1/\sqrt n } ,p(x) = \Pi _{k = 1}^{n - 1} (a^k + x),r_n (x) = x\frac{{p(x) - p( - x)}} {{p(x) + p( - x)}}$$ \end{document}
. The present note gives the asymptotoc formula of max
\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} $$\mathop {\max }\limits_{|x| \leqq 1} \left| {|x| - r_n (x)} \right|$$ \end{document}
.
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