  The Fourier coefficients F(t) of a function f on a compact symmetric space
U/K are given by integration of f against matrix coefficients of irreducible
representations of U. The coefficients depend on a spectral parameter t, which
determines the representation, and they can be represented by elements F(t) in
a common Hilbert space H.
  We obtain a theorem of Paley-Wiener type which describes the size of the
support of f by means of the exponential type of a holomorphic H-valued
extension of F, provided f is K-finite and of sufficiently small support. The
result was obtained previously for K-invariant functions, to which case we
reduce.
