On concentrating idempotents, a survey

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 idempotents 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 wonderful sons. On the mathematical side, one of the best things was discussions with Mischa Zafran concerning a question about linear operators on L (T). 1. From Operators on L (Z) to Concentration 1.1. De…nitions. By L (Z) = ` we mean sequences C = f: : : ; c 1; c0; c1; : : : g of complex numbers such that P jc j < 1. We identify the sequence with its Fourier series C (x) = P c e 2 i . A characteristic function associated to the …nite subset S = fn1; n2; : : : ; nKg of Z is a sequence f: : : ; s 1; s0; s1; : : : g where s = 1 when 2 S and sv = 0 when = 2 S. The product of C = fcng and D = fdng is CD = fcndng, while the convolution is C D = P1 = 1 cn d . A …xed sequence K = fk g creates an operator on functions on Z according to the rule T : C ! K C. We identify TC with the function K (x)C (x) where K (x) = P1 n= 1 kne 2 . By Plancherel’s formula, we have kTCk22 = 1 X n= 1 1 X = 1 kn c 2 = Z 1