Fourier analysis for compact groups

Fourier analysis for compact groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia


Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 1 April 2010

Fourier analysis for compact groups

A function f:G is

  1. representative if there is a finite dimensional representation V of G and vectors v,wV such that f g = v gw for all gG.
  2. square integrable if f 2 2 = G f g f g dμ g <.
  3. smooth if all derivatives exist.
  4. real analytic if f has a power series expansion at each point. C G rep = representative functions  f:G , L 2 G = square integrable functions  f:G , C G = smooth functions  f:G , C ω G = real analytic functions  f:G .

We have a map λ G ^ M d λ functions  f:G.

The set G ^ has a norm .: G ^ 0 . For f ^ λ λ G ^ M d λ define

  1. f ^ λ is finite if all but a finite number of the blocks f ^ λ in f ^ λ are 0,
  2. f ^ λ is square summable if λ G ^ 1 d λ f λ 2 <.
  3. f ^ λ is rapidly decreasing if, for all k >0 , λ k f ^ λ | λ G ^ is bounded,
  4. f ^ λ is exponentially decreasing if, for some K >1 , K λ f ^ λ | λ G ^ is bounded.

Under the map functions  f:G λ G ^ C G rep finite   f ^ λ L 2 Gμ square summable   f ^ λ C G rapidly decreasing   f ^ λ C ω G exponentially decreasing   f ^ λ

The space C G rep is dense in C G and C G L 2 G . In fact the sup norm on C G is related to the L 2 norm on L 2 G and C G is dense in L 2 G .

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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