Open Access
2009/2010 How to Concentrate Idempotents
J. Marshall Ash
Real Anal. Exchange 35(1): 1-20 (2009/2010).

Abstract

Call a sum of exponentials of the form $f(x)=\exp\left( 2\pi iN_{1}x\right) +\exp\left( 2\pi iN_{2}x\right) +\cdot\cdot\cdot+\exp\left( 2\pi iN_{m}x\right) $, where the $N_{k}$ are distinct integers, an idempotent. We have $L^{p}$ interval concentration if there is a positive constant $a$, depending only on $p$, such that for each interval $I\subset\left[ 0,1\right] $ there is an idempotent $f$ so that $\int _{I}\left\vert f\left( x\right) \right\vert ^{p}dx\diagup\int_{0} ^{1}\left\vert f\left( x\right) \right\vert ^{p}dx>a$. We will explain how to produce such concentration for each $p>0$. The origin of this question and the history of the development of its solution will be surveyed.

Citation

Download Citation

J. Marshall Ash. "How to Concentrate Idempotents." Real Anal. Exchange 35 (1) 1 - 20, 2009/2010.

Information

Published: 2009/2010
First available in Project Euclid: 27 April 2010

zbMATH: 1202.42006
MathSciNet: MR2657284

Subjects:
Primary: 42-02 , 42A05 , 47B37
Secondary: 26A05 , 42B99 , 46B10 , 47B34

Keywords: $L^p$ norms , Concentration , Dirichlet kernel , idempotents , projections , restricted type (2,2) , trigonometric polynomials , weak type (2,2)

Rights: Copyright © 2009 Michigan State University Press

Vol.35 • No. 1 • 2009/2010
Back to Top