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)