An interference problem for exponentials
type: Item
instance of: scholarly article
title: An interference problem for exponentials
zbMATH Open document ID: 0682.30017
author: B. F. Logan
published in: Michigan Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The problem (posed by H. L. Montgomery) is to determine the supremum C(T) of \[ \int^{T}_{-T}| B(t)|^ 2 dt/\int^{T}_{-T}| A(t)|^ 2dt \] over \(A(t)=\sum^{N}_{k=1}a_ k \exp (i\lambda_ kt)\), \(a_ k>0\), and \(B(t)=\sum^{N}_{k=1}b_ k \exp (i\lambda_ kt)\), where \(| b_ k| \leq a_ k\). This paper discusses several aspects of this and related problems. In particular, the author obtains both upper and lower bounds for C(T). The methods are both ingenious and powerful, but too complicated to be described in detail here. An interesting application is the (best possible) inequality \[ (2T)^{- 1}\int^{T}_{-T}| \sum^{\infty}_{k=-\infty}a_ k \exp (i\lambda_ kt)|^ 2 dt\geq \sum a^ 2_ k, \] where \(a_ k\geq 0\).
Mathematics Subject Classification ID: 30D10
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 4116885
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003818
OpenAlex ID: W2060453813
DOI: 10.1307/MMJ/1029003818
description: scientific article; zbMATH DE number 4116885