152 research outputs found
Turan's method and compressive sampling
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
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