Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.

Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.