An estimate of theL 1-norm of an exponential sum

descriptionPublicationkeyboard_double_arrow_right Article 01 Sep 1996 English Publisher:Springer Science and Business Media LLCJournal:Mathematical Notes, volume 64, pages 401-404 (issn: 0001-4346, eissn: 1573-8876,

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Authors: Karatsuba, A. A.;

doi: 10.1007/bf02314851

An estimate of theL 1-norm of an exponential sum

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Abstract

The following theorem is proved. Suppose that \(A\geq 1/2\), \(10\) the inequality \[ \int_0^1 \Biggl| \sum_{n=1}^N \alpha_n \exp(2\pi ixf(n)) \Biggr| dx\geq \exp(2^{-15} A^{-2} (\log N)^{3-2\beta}) \] is valid. This improves an estimate of S. V. Bochkarev. The proof uses an estimate of a trigonometric sum due to O. V. Popov.

Related Organizations

Steklov Mathematical Institute
Russian Federation
Department of Mathematical Sciences
Russian Federation
Russian Academy of Sciences
Russian Federation

Keywords

estimate, Trigonometric polynomials, inequalities, extremal problems, \(L_1\)-norm of an exponential sum, Estimates on exponential sums

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