On the Notion of Bandlimitedness and its Generalizations

in this survey article we introduce the paley-wiener space of bandlimited functions and review some of its generalizations. some of these generalizations are new and will be presented without proof because the proofs will be published somewhere else. guided by the role that the differentiation operator plays in some of the characterizations of the paley-wiener space, we construct a subspace of vectors in a hilbert space using a self-adjoint operator we then show that the space has similar properties to those of the space the paper is concluded with an application to show how to apply the abstract results to integral transforms associated with singular sturm-liouville problems.