×

Use of complex analysis for deriving lower bounds for trigonometric polynomials. (English. Russian original) Zbl 0914.42001

Math. Notes 63, No. 6, 709-716 (1998); translation from Mat. Zametki 63, No. 6, 803-811 (1998).
Summary: It is shown that for any distinct natural numbers \(k_1,\dots, k_n\) and arbitrary real numbers \(a_1,\dots, a_n\) the following inequality holds: \[ -\min_x \sum^n_{j= 1} a_j(\cos(k_jx)- \sin(k_jx))\geq B\Biggl({1\over 1+\ln n} \sum^n_{j= 1} a^2_j\Biggr)^{1/2},\qquad n\in\mathbb{N}, \] where \(B\) is a positive absolute constant (for example, \(B= 1/8\)). An example shows that in this inequality the order with respect to \(n\), 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.

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
11L03 Trigonometric and exponential sums (general theory)
30D55 \(H^p\)-classes (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. K. Pichorides, ”A remark on exponential sums,”Bull. Amer. Math. Soc.,83, 283–285 (1977). · Zbl 0346.42001
[2] A. Zygmund,Trigonometric Series, Vol. 1, 2, Cambridge Univ. Press, Cambridge (1959, 1960); Russian translation: Mir, Moscow (1965). · Zbl 0085.05601
[3] N. K. Bari,Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961).
[4] P. Koosis,Introduction to H p -Spaces Cambridge Univ. Press, Cambridge (1980). · Zbl 0435.30001
[5] S. V. Konyagin, ”On a problem of Littlewood,”Mat. Zametki [Math. Notes],49, No. 2, 143–144 (1991). · Zbl 0776.42001
[6] S. A. Pichugov, ”Estimates of the minimum of trigonometric sums,” in: ”Studies of Modern Summation and Function Approximation Problems and Their Applications [in Russian], Dnepropetrovsk State Univ., Dnepropetrovsk (1982), pp. 35–38.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.