Jam8 soumis a Topics on the Interface between Harmonic Analysis and Number Theory T Erdelyi B Saffari G Tenenbaum Eds

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Niveau: Supérieur, Licence, Bac+2
[Jam8] soumis a Topics on the Interface between Harmonic Analysis and Number Theory, T. Erdelyi, B. Saffari, G. Tenenbaum (Eds). The phase retrieval problem for cyclotomic crystals Philippe JAMING Abstract : In this survey, we present the results on the phase retrieval prob- lem for cyclotomic crystals, following J. Rosenblatt's paper [Ro] in a simplified setting. We then present some extensions to the triple-correlation function du to the author and M. Kolountzakis [JamK] and conclude with some open problems. Keywords : phase retrieval problem. AMS subject class : 48A85, 58G35. 1. Introduction 1.1. Phase retrieval problems. Usually, when one measures a quantity, due to noise, poor measurement equipment, transmi- tion in messy media... the phase of the quantity one wishes to know is lost. In mathematical terms, one wants to know a quantity ?(t) knowing only |?(t)| for all t ? Rd. Stated as this, the problem has too many solutions to be useful and one tries to incorporate a priori knowledge on ? to decrease the underterminancy. A typical situation is that ? = f? for some compactly supported function f ? L2(Rd) or more generally for some compactly supported Schwartz distribution f ? S ?(Rd). Let us temporarilly concentrate on the one-dimensional case for finite-energy signals.

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Publié par

profil-vieg-2012

Langue

English

[Jam8]

soumis`aTopicsontheInterfacebetweenHarmonicAnalysisandNumberTheory, T. Erdelyi, B. Saﬀari, G. Tenenbaum (Eds). The phase retrieval problem for cyclotomic crystals Philippe JAMING Abstract : In this survey, we present the results on the phase retrieval prob-lem for cyclotomic crystals, following J. Rosenblatt’s paper [Ro] in a simpliﬁed setting. We then present some extensions to the triple-correlation function du to the author and M. Kolountzakis [JamK] and conclude with some open problems. Keywords : phase retrieval problem. AMS subject class : 48A85, 58G35. 1. Introduction 1.1. Phase retrieval problems. Usually, when one measures a quantity, due to noise, poor measurement equipment, transmi-tion in messy media... the phase of the quantity one wishes to know is lost. In mathematical terms, one wants to know a quantity ϕ ( t ) knowing only | ϕ ( t ) | for all t ∈ R d . Stated as this, the problem has too many solutions to be useful and one tries to incorporate a priori knowledge on ϕ to decrease the underterminancy. A typical situation is that ϕ = f b for some compactly supported function f ∈ L 2 ( R d ) or more generally for some compactly supported Schwartz distribution f ∈ S 0 ( R d ). Let us temporarilly concentrate on the one-dimensional case for ﬁnite-energy signals. The problem is then: Problem 1. Given f ∈ L c 2 ( R ) , ﬁnd all g ∈ L c 2 ( R ) such that | f b ( ξ ) | = | g b ( ξ ) | for (almost) all ξ ∈ R . b As f is compactly supported, its Fourier transform is analytic so that actually | f ( ξ ) | = | g b ( ξ ) | for all ξ ∈ R . Problem 1 has been solved explicitly by Walter [Wa] and we may now describe this solution. First note that this problem has trivial solutions g ( t ) = cf ( t − α ) and g ( t ) = cf ( − t − α ) where c ∈ C with | c | = 1 and α ∈ R . However, there may be more solutions: as f ∈ L c 2 ( R ), b with support [ − σ, σ ], then, f is an entire function of order 1 of type σ . By Hadamard’s Factorization Theorem, one may then write f b ( z ) = z k e az + b Y  1 − zz k  e z/z k b where a, b ∈ R and the z k ’s are the zeroes of f in the complex plane. Moreover, this zeroes b b essentially characterize f , so that if we knew | f ( ξ ) | for all ξ ∈ C (and not only ξ ∈ R ) we would be done. b b To overcome this, write | f b ( ξ ) | 2 = | g b ( ξ ) | 2 as f ( ξ ) f ( ξ ) = b g ( ξ ) g b ( ξ ) and note that this equation b extends to ξ in all of C . It follows that a zero of g b is either a zero of f or a conjugate of a zero 321

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