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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