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483operators versus groups

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  • antianticamper
    Feb 2, 2011
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      Hello all,

      Harmonic analysis is not my specialty so please forgive me if my
      questions are naive. Years ago I studied quantum algorithms and
      understood them by learning about harmonic analysis over finite groups,
      mostly finite Abelian but also some finite, non-commutative groups. I
      studied the more general harmonic theory (locally compact groups, etc.)
      a bit but not too extensively. Recently, I learned about harmonic
      analysis on manifolds as the study of eigenfunctions of the
      Laplace-Beltrami operator, which was unknown to me and on the surface a
      different approach from the group-theoretic approach. Here are my
      questions:

      1. Spherical harmonics can be derived as the eigenfunctions of the
      Laplacian operator on a sphere OR via the transitive action of SO(3) on
      the sphere as a homogeneous space. What is the essential connection
      between these two approaches to harmonic analysis? How can I see these
      derivations as being "the same"?

      2. The motivation for my current study is to compute the spectrum of
      the Laplace-Beltrami operator on 2-manifolds for intrinsic shape
      characterization, sometimes called shape-DNA, for example on brain
      surfaces. What other operators may also be useful in this way for this
      purpose?

      3. Any references to nice introductions to operator spectra on
      low-dimensional manifolds would be greatly appreciated, (especially with
      my computational application in view)!

      aac
      (Medical Imaging Informatics - UCLA)