Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel

Under the assumption of positive multiplicity, we obtain basic estimates of the hypergeometric functions F_\la and G_\la of Heckman and Opdam, and sharp estimates of the particular functions F_0 and G_0. Next we prove the Paley-Wiener theorem for the Schwartz class, solve the heat equation and estimate the heat kernel.