In this dissertation, we study sharp boundedness of the Bochner-Riesz means and spectral projection operators for the Hermite and special Hermite expansions. Looking into these problems in ways that have not been previously attempted, we improve known results. We first consider the problem of estimating optimal bounds on the L^p--L^q operator norms of the associated spectral projection operators for general p, q. These estimates have been mainly studied when p or q is 2. We establish various new sharp estimates in an extended range of p, q. As an application, we obtain new results on the uniform resolvent estimate and the Carleman inequality for the heat operator. We also investigate boundedness properties of the Bochner-Riesz means for the Hermite and special Hermite expansions. Concerning this issue, the previous works have been based on a common strategy utilizing the spectral projection estimates as a key ingredient. However, such a strategy has a technical limit, so far there have been no sharp results beyond a class of Lebesgue spaces given by the Tomas-Stein theorem. We introduce a new approach to this problem which employs recent results for oscillatory integral operators.