Is There a Generalized Fourier Transform for All Manifolds?

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SUMMARY

The discussion centers on the existence of a generalized Fourier transform applicable to all manifolds, positing that the Fourier transform in Euclidean space is a specific instance of this broader concept. Participants explore the implications of using eigenfunctions of the Laplace-Beltrami operator, defined as ##\nabla^2 = \nabla_a \nabla^a = g^{ab} \nabla_a \nabla_b##, for expansions in Riemannian and pseudo-Riemannian manifolds. The conversation highlights the need for a unified framework that extends traditional Fourier analysis to more complex geometric structures.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with the Laplace-Beltrami operator
  • Knowledge of eigenfunction expansions
  • Basic principles of Fourier analysis
NEXT STEPS
  • Research the properties of the Laplace-Beltrami operator in various manifolds
  • Explore the concept of eigenfunction expansions in Riemannian geometry
  • Study existing generalized Fourier transforms in mathematical literature
  • Investigate applications of Fourier transforms in physics and engineering on manifolds
USEFUL FOR

Mathematicians, theoretical physicists, and researchers in differential geometry seeking to extend Fourier analysis to complex manifolds.

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Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?
 
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The corresponding thing in a (Riemannian or pseudo-Riemannian) manifold would be to look for expansions in terms of eigenfunctions of the operator ##\nabla^2 = \nabla_a \nabla^a = g^{ab} \nabla_a \nabla_b##.
 

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