Turan's method and compressive sampling

TL;DR

This paper explores Turan's method for constructing sequences with small trigonometric polynomials, improves bounds on their size using probabilistic and explicit methods, and connects these results to signal recovery in cyclic groups.

Contribution

It provides improved estimates for the size of frequency sets in Turan's problem and introduces explicit constructions, extending the analysis to cyclic groups relevant for signal processing.

Findings

Improved bounds on M(n, d) for Turan's problem.

Explicit constructions of frequency sequences with desired properties.

Application of probabilistic methods to signal recovery in cyclic groups.

Abstract

Turan's method, as expressed in his books, is a careful study of trigonometric polynomials from different points of view. The present article starts from a problem asked by Turan: how to construct a sequence of real numbers x(j) (j= 1,2,...n) such that the almost periodic polynomial whose frequencies are the x(j) and the coefficients are 1 are small (say, their absolute values are less than n d, d< given) for all integral values of the variable m between 1 and M= M(n,d) ? The best known answer is a random choice of the x(j) modulo 1. Using the random choice as Turan (and before him Erd\"os and Renyi), we improve the estimate of M (n, d) and we discuss an explicit construction derived from another chapter of Turan's book. The main part of the paper deals with the corresponding problem when R / Z is replaced by Z / NZ, N prime, and m takes all integral values modulo 1 except 0. Then it has an interpretation in signal theory, when a signal is representad by a function on the cyclic goup G = Z / NZ and the frequencies by the dual cyclic group G^ : knowing that the signal is carried by T points, to evaluate the probability that a random choice of a set W of frequencies allows to recover the signal x from the restriction of its Fourier tranform to W by the process of minimal extrapolation in the Wiener algebra of G^(process of Cand\`es, Romberg and Tao). Some random choices were considered in the original article of CRT and the corresponding probabilities were estimated in two preceding papers of mine. Here we have another random choice, associated with occupancy problems.