An estimate of the \(L_1\)-norm of an exponential sum
type: Item
instance of: scholarly article
title: An estimate of the \(L_1\)-norm of an exponential sum
zbMATH Open document ID: 0924.11065
DOI: 10.1007/BF02314851
published in: Mathematical Notes
publication date: 1999-11-15T00:00:00Z
review text: The following theorem is proved. Suppose that \(A\geq 1/2\), \(1<\beta< 3/2\), and \(f(n)= [e^{A(\log n)^\beta}]\). Then for any coefficients \(\alpha_n= \pm 1\) and for \(N\geq N_1(\beta,A)>0\) 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.
reviewed by: Dieter Leitmann
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 42A05
zbMATH DE Number: 1294981
zbMATH Keywords: \(L_1\)-norm of an exponential sum
zbMATH Keywords: estimate
author: Anatolii A. Karatsuba
MaRDI profile type: Publication
cites work: Q3254471
cites work: Hardy's inequality and the \(L^ 1\) norm of exponential sums
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description: scientific article; zbMATH DE number 1294981