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Inverse-constrained projection filters

This paper describes the use of moving-block expansions in Slepian sequences (discrete prolate spheroidal sequences) as a method to generate precision complex demodulators for data analysis. Such filters are used to separate the annual cycle from the low-frequency trend in climate date, to isolate individual modes in helioseismology data, etc., so statistical efficiency and reliability is required. In particular one must combine the conflicting requirements of isolating a narrow frequency band with having only a short data series available. For such uses, the concentration properties of the Slepian sequences give optimum protection against signals at frequencies other than the band of interest. However, like other orthogonal expansions, the standard inverse suffers from Gibb's ripples and similar effects. Here, the usual orthogonal series expansion is replaced with a partial inverse-theory reconstruction with a smoothness constraint. Defining a set of polynomials orthogonal with respect to inner products with the Slepian sequences allows construction of a sequence of projection operators with variable smoothness properties.