On Slepian series expansion for digitized signals (with X. Shen), submitted for publication, 2010.
Prolate spheroidal wave functions (Slepian functions) are special functions
that are most localized in both spatial and frequency domain, simultaneously.
They lead to the optimal solution of the concentration problem once posed by
Claude E. Shannon. This fact was unraveled by David Slepian and his
collaborators at Bell Lab in 1960s. Since then this system has shown promise
for many applications in engineering and some other areas. Unlike usual
orthogonal polynomials or trigonometric systems, Slepian functions possess
peculiar properties, such as, dual orthogonality, duality of time-frequency
representation, and multiscale structure, to name a few. This paper is devoted
to the study of Slepian series for digitized functions in the Paley-Wiener
space and beyond. We shall give the convergence analysis of the expansion
coefficients and explore their properties by numerical experiments.
We conclude the paper by discussing problems raised in such expansions used in
the practice when only the discrete data are available and contaminated by noise.
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