Convergence of generalized eigenfunction expansions

Abstract

We present a simplified theory of generalized eigenfunction expansions for a commuting family of bounded operators and with finitely many unbounded operators. We also study the convergence of these expansions, giving an abstract type of uniform convergence result, and illustrate the theory by giving two examples: The Fourier transform on Hecke operators, and the Laplacian operators in hyperbolic spaces.

Keywords

• Generalized eigenfunction expansion Generalized eigenprojection
• Fourier transform
• differential operators
• Hecke operators
• modular group.