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Mat. Zametki, 1998, Volume 63, Issue 6, Pages 803–811 (Mi mz1350)  

Use of complex analysis for deriving lower bounds for trigonometric polynomials

A. S. Belov

Ivanovo State University

Abstract: It is shown that for any distinct natural numbers $k_1,…,k_n$ and arbitrary real numbers $a_1,…,a_n$ the following inequality holds:
$$ -\min_x\sum_{j=1}^na_j(\cos(k_jx)-\sin(k_jx)) \ge B(\frac 1{1+\ln n}\sum_{j=1}^na_j^2)^{1/2}, \qquad n\in\mathbb N, $$
where $B$ is a positive absolute constant (for example, $B=1/8$). An example shows that in this inequality the order with respect ton, i.e., the factor $(1+\ln n)^{-1/2}$, cannot be improved. A more elegant analog of Pichorides' inequality and some other lower bounds for trigonometric sums have been obtained.

DOI: https://doi.org/10.4213/mzm1350

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English version:
Mathematical Notes, 1998, 63:6, 709–716

Bibliographic databases:

UDC: 517.5
Received: 12.04.1997

Citation: A. S. Belov, “Use of complex analysis for deriving lower bounds for trigonometric polynomials”, Mat. Zametki, 63:6 (1998), 803–811; Math. Notes, 63:6 (1998), 709–716

Citation in format AMSBIB
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\paper Use of complex analysis for deriving lower bounds for trigonometric polynomials
\jour Mat. Zametki
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\vol 63
\issue 6
\pages 803--811
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\jour Math. Notes
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\vol 63
\issue 6
\pages 709--716
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