Uniqueness theorems for Fourier transforms

Nir Lev

doi:10.1016/j.bulsci.2010.12.002

Uniqueness theorems for Fourier transforms

Nir Lev

Department of Mathematics

Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

Abstract

Let Γ be a smooth curve in the plane R², and μ be any subset of R². When can one recover uniquely a finite measure Γ supported by Γ and absolutely continuous with respect to the arc length measure on Λ from the restriction to Λof its Fourier transform? In this note we present two results in the subject, one is concerned with the case when Γ is a circle, and the other with the case when Γ is "close" to a lattice.

Original language	English
Pages (from-to)	134-140
Number of pages	7
Journal	Bulletin des Sciences Mathematiques
Volume	135
Issue number	2
DOIs	https://doi.org/10.1016/j.bulsci.2010.12.002
State	Published - Mar 2011

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1016/j.bulsci.2010.12.002

Cite this

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title = "Uniqueness theorems for Fourier transforms",

abstract = "Let Γ be a smooth curve in the plane R2, and μ be any subset of R2. When can one recover uniquely a finite measure Γ supported by Γ and absolutely continuous with respect to the arc length measure on Λ from the restriction to Λof its Fourier transform? In this note we present two results in the subject, one is concerned with the case when Γ is a circle, and the other with the case when Γ is {"}close{"} to a lattice.",

author = "Nir Lev",

note = "Funding Information: The latter condition implies, however, that if ν is non-zero then the closed support of ν is a set of infinite 1-dimensional Hausdorff measure (see [2, pp. 132–134]). But this is not possible since ν is supported by the image of the curve Γ under the projection R2 → T2. Hence ν = 0.",

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N2 - Let Γ be a smooth curve in the plane R2, and μ be any subset of R2. When can one recover uniquely a finite measure Γ supported by Γ and absolutely continuous with respect to the arc length measure on Λ from the restriction to Λof its Fourier transform? In this note we present two results in the subject, one is concerned with the case when Γ is a circle, and the other with the case when Γ is "close" to a lattice.

AB - Let Γ be a smooth curve in the plane R2, and μ be any subset of R2. When can one recover uniquely a finite measure Γ supported by Γ and absolutely continuous with respect to the arc length measure on Λ from the restriction to Λof its Fourier transform? In this note we present two results in the subject, one is concerned with the case when Γ is a circle, and the other with the case when Γ is "close" to a lattice.

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DO - 10.1016/j.bulsci.2010.12.002

M3 - مقالة

SN - 0007-4497

VL - 135

SP - 134

EP - 140

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

IS - 2

ER -

Uniqueness theorems for Fourier transforms

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