Estimate of fractional moments of a trigonometric sum
type: Item
instance of: scholarly article
title: Estimate of fractional moments of a trigonometric sum
zbMATH Open document ID: 1161.44307
DOI: 10.1023/B:MATN.0000036767.14036.76
published in: Mathematical Notes
publication date: 2004-12-07T00:00:00Z
review text: Suppose that \(n = \sum_{k=0}^\infty \varepsilon_k2^k\), where \(\varepsilon_k = 0, 1\), is the binary representation of positive integers \(n\). Split the set of positive integers into two nonintersecting classes as follows: \(\mathbb N_0 =\left\{n: n \in \mathbb N,\;\sum_{k=0}^\infty \varepsilon_k\equiv 0\pmod 2\right\}\) and \(\mathbb N_1 =\left\{n: n \in \mathbb N,\;\sum_{k=0}^\infty \varepsilon_k\equiv 1\pmod 2\right\}\). Define the trigonometric sum \[ S(\alpha) =\sum_{n=1}^{2^{N+1}-1} \varepsilon_k e^{2\pi i\alpha n} =\prod_{r=0}^{N-1}\left(1-e^{2\pi i\alpha 2^r}\right). \] In this paper, the authors give an upper bound for the fractional moment of the sum \(S(\alpha)\). In addition, they obtain estimates and an asymptotic formula for fractional moments of a trigonometric sum closely related to \(S(\alpha)\). Theorem. Suppose that \(\varepsilon\) is an arbitrary number such that \(0 < \varepsilon < 0.01\) and \(k\) is a positive integer, \(k > \log_2([1/\varepsilon] + 1)\). Then the following estimate holds: \[ \int_0^1 | S(\alpha)|^{1/k}\, d\alpha \leq \left(\frac 4\pi (1+\varepsilon)\right)^{N/k+1} \] .
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 2121468
zbMATH Keywords: fractional moment
zbMATH Keywords: trigonometric sum
zbMATH Keywords: coprime integers
zbMATH Keywords: Fourier series
author: K. M. Èminyan
MaRDI profile type: Publication
full work available at URL: b:matn.0000036767.14036.76
OpenAlex ID: W2012751139
description: scientific article; zbMATH DE number 2121468