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Pompeiu's problem on discrete space.

D Zeilberger1

  • 1The Institute for Advanced Study, Princeton, New Jersey 08540.

Proceedings of the National Academy of Sciences of the United States of America
|August 1, 1978
PubMed
Summary

This study investigates the Pompeiu problem for finite families of subsets in n-dimensional lattices. It determines conditions under which only the zero function satisfies specific integral equations.

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Area of Science:

  • Harmonic Analysis
  • Geometric Measure Theory
  • Multidimensional Analysis

Background:

  • The Pompeiu problem traditionally examines unique function properties based on geometric shapes.
  • Extending this problem to higher dimensions and families of sets presents unique mathematical challenges.
  • Understanding function behavior on lattices is crucial in various fields, including signal processing and physics.

Purpose of the Study:

  • To generalize the Pompeiu problem to finite families of finite subsets within the n-dimensional lattice Z(n).
  • To identify the precise conditions on these families of subsets that force a function to be identically zero.
  • To analyze the implications of translation invariance within the lattice group tau.

Main Methods:

  • The study employs techniques from Fourier analysis and spectral theory.
  • It involves analyzing the properties of specific integral operators defined over the lattice Z(n).
  • Methods include investigating the null spaces of these operators under varying family configurations.

Main Results:

  • Characterization of families of subsets for which the Pompeiu problem has only the trivial solution.
  • Identification of key geometric or combinatorial properties of the subsets that dictate the solution.
  • The results provide necessary and sufficient conditions for the zero function to be the unique solution.

Conclusions:

  • The Pompeiu problem on n-dimensional lattices is solvable and depends critically on the geometric configuration of the subset family.
  • The findings contribute to the understanding of unique continuation properties for functions on discrete spaces.
  • This research offers a foundation for further exploration of integral equations and function analysis in higher-dimensional lattices.

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