Prolate spheroidal wave functions, an introduction to the Slepian series and its properties |
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Authors: | Ian C Moore Michael Cada |
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Institution: | a Queens University, Department of Mathematics and Statistics, Kingston, Ontario, Canada K7L 3N6;b University of Ottawa, School of Information Technology and Engineering (SITE), Ottawa, Ontario, Canada K1N 6N5;c Dalhousie University, Department of Computer and Electrical Engineering, Halifax, Nova Scotia, Canada B3J 1Z1 |
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Abstract: | For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter. |
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Keywords: | Interpolation Orthogonal expansion Prolate spheroidal wave function |
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