Uncertainty Principles, Prolate Spheroidal Wave Functions, and Applications

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In the literature, the prolate spheroidal wave functions (PSWFs) are often regarded as mysterious set of functions of L2(), with no explicit or standard representation and too difficult to compute numerically. Nonetheless, the PSWFs exhibit the unique properties to form an orthogonal basis of L2([ − 1, 1]), an orthonormal system of L2(R) and an orthonormal basis of B c , the Paley-Wiener space of c − band-limited functions. Recently, there is a growing interest in the computational side of the PSWFs as well as in the applications of these laters in solving many problems from different scientific area, such as physics, signal processing and applied mathematics. In this work, we first give a brief description of the main properties of the PSWFs. Then, we give a detailed study of our two recent computational methods of these PSWFs and their associated eigenvalues. Also, we give a brief description for a composite quadrature based method for the approximation of the values and the eigenvalues of the high frequency PSWFs. In the applications part of this work, we study the quality of approximation by the PSWFs in the space of almost band-limited functions. Moreover, we study the contribution of the PSWFs in the reconstruction of band-limited functions with missing data sets. Finally, we provide the reader with some numerical examples that illustrate the results of this work.


Orthogonal Polynomial Uncertainty Principle Legendre Polynomial Quadrature Method Schmidt Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences of BizerteUniversity of CarthageJarzounaTunisia

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