No characterization of generators in ℓ^p(1 < p < 2) by zero set of Fourier transform

Given 1 < p < 2 we construct two continuous functions f and g on the circle, with the following properties:. (i) They have the same set of zeros;. (ii) The Fourier transforms over(f, ̂) and over(g, ̂) both belong to ℓ^p (Z);. (iii) The translates of over(g, ̂) span the whole ℓ^p, but those of over(f, ̂) do not. A similar result is true for L^p (R). This should be contrasted with the Wiener theorems related to p = 1, 2. To cite this article: N. Lev, A. Olevskii, C. R. Acad. Sci. Paris, Ser. I 346 (2008).