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New York Journal of Mathematics 9 (2003), 149-164.

Reconstruction of functions from their triple correlations

Philippe Jaming and Mihail N. Kolountzakis

Published: October 23, 2003
Keywords: k-deck; phase retrieval; bispectrum; triple correlation
Subject: 42A99; 94A12

Abstract:

Suppose that $A$ is a subset of an abelian group $G$. To know the {\em $3$-deck} of $A$ is to know the number of occurrences in $A$ of translates of each possible multiset $\{0,a,b\}$. The concept of the 3-deck of a set is naturally extended to $L^1$ functions on $G$. In this paper we study when the 3-deck of a function determines the function up to translations. The method is to look at the Fourier Transform of the function. Our emphasis is on the real line and the cyclic groups.

Acknowledgments:
Research partially financed by: {\it European Commission} Harmonic Analysis and Related Problems 2002-2006 IHP Network (Contract Number: HPRN-CT-2001-00273 - HARP)

Author information:
Philippe Jaming :
Universit\'e d'Orl\'eans, Facult\'e des Sciences, D\'epartement de Ma\-th\'e\-ma\-ti\-ques, BP 6759, F 45067 Orl\'eans Cedex 2, France
jaming@labomath.univ-orleans.fr
http://www.univ-orleans.fr/SCIENCES/MAPMO/membres/jaming/

Mihail N. Kolountzakis:
Department of Mathematics, University of Crete, Knossos Ave., 714 09 Iraklio, Greece
kolount@member.ams.org
http://fourier.math.uoc.gr/~mk/