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Determining a distribution from the modulus of its fourier transform
Abstract
Let D be a tempered distribution on the real line R The problem of determining all tempered distributions E on R with |Ecirc;|=|[Dcirc]| is studied for distributions with compact support If there exists A.B tempered distributions with compact support such that D= AB and E=AB then |Ê|=|[Dcirc]|, however, it is shown that there are smooth functions D. E with compact support such that |Ê|=|[Dcirc]|, but for which there is no such factorization. This requires having some knowledge of the distribution of the zeros of [Dcirc]: and partly to this end, generalizations of theorems of Titchmarsh on the distribution of the zeros of a Fourier transform are proved.
... En [33], Rosenblatt estudia el caso de distribuciones más generales. ...
Tesis doctoral inédita de la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 15-12-1997 Bibliografía: h. 68-71
We show that a σ-finite Borel measure, which is defined on the positive half-line and has a nonincreasing density (with respect to the Lebesgue measure) is uniquely characterized by its symmetrization.
IfX is a locally compact abelian group, [^(m)]\hat \mu
its Fourier transform, the mapping [^(m)]\hat \mu
| is obviously not injective. The aim of this article is to find conditions under which the identification of [^(m)]\hat \mu
| is possible up to a shift and a central symmetry.
It is generally asserted that valuable phase information is irretrievably lost in the squaring operation required to obtain the intensity from the amplitude distribution in the image of a star formed by a lens. In this paper the phase reconstruction is studied under the constraint that the aperture of the lens be finite. It is shown that in some cases the phase reconstruction is unique; and that in other cases there is finite or denumerable infinite number of discrete solutions. Conversely, it is proved that every non-negative function that is integrable and band-limited can be realized as the intensity distribution in the image of a star formed by a lens.
The problem of phase retrieval arises in experimental uses of diffraction to determine intrinsic structure because the modulus of a Fourier transform is all that can usually be measured after diffraction occurs. For finite distributions, the phase retrieval problem can be solved by methods of factorization in suitable rings of polynomials; for continuous distributions with compact support, the methods of complex analysis are needed to solve the phase retrieval problem. These methods are discussed and examples are given for illustration.
Thhrit. dr.\ Di,~irihirtio~r,s. Hermann. Paris
- L Schwartz
L. Schwartz. Thhrit. dr.\ Di,~irihirtio~r,s. Hermann. Paris. 1966.
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- A Wa1ther
A. Wa1ther.Thequestion orphase retrieval in optics. Opric<r.fcitr 10 11963j.41 49 Downloaded by [Michigan State University] at 05:50 06 January 2015
Analytic Function Theory
- J Rosenblatt