Title: From Double Hecke Algebra to Analysis

We discuss $q$-counterparts of the Gauss integrals, a new type of Gauss-Selberg sums at roots of unity, and $q$-deformations of Riemann's zeta. The paper contains general results, one-dimensional formulas, and remarks about the current projects involving the double affine Hecke algebras.

1991 Mathematics Subject Classification:

Keywords and Phrases: Hecke algebra, Fourier transform, spherical function, Macdonald polynomial, Gauss integral, Gaussian sum, metaplectic representation, Verlinde algebra, braid group, zeta function.

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