A variant on the compressed sensing of Emmanuel Candès Basarab Matei and Yves Meyer

This paper is motivated by the outsanding achievements of Emmanuel Candès and Terence Tao on what is now called “compressed sensing”. Let us begin with a theorem by Terence Tao. Let p be a prime number and Fp be the finite field with p elements. We denote by #E the cardinality of E ⊂ Fp. The Fourier transform of a complex valued function f defined on Fp is denoted by f̂ . Let Mq be the collection of all f : Fp 7→ C such that the cardinality of the support of f does not exceed q. Then Terence Tao proved that for q < p/2 and for any set Ω of frequencies such that #Ω ≥ 2q, the mapping Φ : Mq 7→ l(Ω) defined by f 7→ f̂ is injective. We want to generalize this fact to functions defined on the unit square with applications to image processing. In a forthcoming work the hypothesis that f is supported by the unit square will be removed. Here and in what follows the action takes place on [0, 1] identified to (R/Z). Since the unit square [0, 1] has been identified to (R/Z), the Fourier transform of f ∈ L([0, 1]) is the