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


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.


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


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

zbMATH: 1202.42006
MathSciNet: MR2657284

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
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