  In the paper, we introduce and calculate difference Fourier transforms on
representations of the double affine Hecke algebras in polynomilas, polynomials
multiplied by the Gaussian, and various spaces of delta-functions including
finite-dimensional ones, give a general description of the semisimple
representations with a special consideration of the GL-case, and then gradually
restrict ourselves with spherical, pseudo-unitary, and Fourier-invariant
representations. The latter generalize the Verlinde algebras and lead to new
Gauss-Selberg sums and Macdonald's eta-type identities.
