 
Summary: On concentrating idempotents, a survey
J. Marshall Ash
Abstract. 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 or, simply, an idempotent. It is known that for every
p > 1; and every set S of the torus T = R=Z with jSj > 0; there are idempo
tents concentrated on S in the Lp sense. We sketch how this concentration
phenomenon originated as a reformulation of a functional analysis problem,
and, in turn, studying concentration led to some interesting questions about
Lp norms of Dirichlet kernels associated with multiple trigonometric series.
Some counterexamples involving linear operators not of convolution type are
given.
In 1977 I was visiting Stanford on sabbatical from DePaul. It was a most
productive year, both personally and professionally. One the ...rst side, I met and
married Alison who subsequently gave me the second and third of my three won
derful sons. On the mathematical side, one of the best things was discussions with
Mischa Zafran concerning a question about linear operators on L2
(T).
1. From Operators on L2
(Z) to Concentration
