Composition operators on Paley-Wiener type spaces.

摘要

Sampling theory is an active field of research and it spans various fields from communication engineering to pure mathematics. Shannon's sampling theorems provide algorithms to reconstruct the bandlimited signals from their discrete sampling. In other words, these theories provide the crucial connection between continuous and discrete representations of information that enables one to store continuous signals as discrete, digital data with minimal error.;In this dissertation we have studied extensively the theory of composition operator on various Hilbert spaces, specially reproducing kernel Hilbert spaces associated with positive definite kernels. The celebrated Paley-Wiener theorem naturally identifies the spaces of bandlimited signals with spaces of entire functions of exponential type. In this dissertation, we focused on boundedness of composition and weighted composition operators on these type of spaces. Very recently, it has been shown that these spaces remain invariant only under composition with affine maps. After some motivation demonstrating the importance of characterization of range spaces arising from the action of more general composition operators on the spaces of bandlimited functions, we identified the subspaces of square integrable functions L² (R) generated by these actions. Extensions of these theorems where Paley-Wiener spaces are replaced by more general de Branges-Rovnyak spaces are given.;We have also tried to develop a correspondence between general filters in signal processing and the filters that arise from diffeomorphisms of the underlying domain.