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Digital filtering and prolate functions

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A class of trigonometric polynomialsp(x) = sum_{n=-N}^{N} a_{n} e^{j n pi x}of unit energy is introduced such that their energy concentrationalpha = int_{-e}^{e} p^{2}(x) dxin a specified interval(- epsilon, epsilon)is maximum. It is shown that the coefficientsa_{n}must be the eigenvectors of the systemsum_{m=-N}^{N} frac{sin (n - m)pi epsilon}{(n - m)epsilon} a_{m} = lambda a_{n}. corresponding to the maximum eigenvalue X. These polynomials are determined forN = 1, cdots , 10andepsilon = 0.025, cdots , 0.5. The resulting family of periodic functions forms the discrete version of the familiar prolate spheroidal wave functions.

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Circuit Theory, IEEE Transactions on  (Volume:19 ,  Issue: 6 )