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-lambda and G(lambda) 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.