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Exponential sums with coe cients 0 or 1 and concentrated Lp

Summary: Exponential sums with coe cients 0 or 1 and
concentrated Lp
Bruce Anderson J. Marshall Ash Roger Jonesy
Daniel G. Rider Bahman Sa ari
May 4, 2007
A sum of exponentials of the form f(x) = exp (2 iN1x)+exp (2 iN2x)+
+ exp (2 iNmx), where the Nk are distinct integers is called an
idempotent trigonometric polynomial (because the convolution of f
with itself is f) or, simply, an idempotent. We show that for every
p > 1; and every set E of the torus T = R=Z with jEj > 0; there are
idempotents concentrated on E in the Lp sense. More precisely, for
each p > 1; there is an explicitly calculated constant Cp > 0 so that
for each E with jEj > 0 and > 0 one can nd an idempotent f
such that the ratio
E jfjp R
T jfjp 1=p
is greater than Cp . This


Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University


Collections: Mathematics