Uniform Estimates of the Prolate Spheroidal Wave Functions and Spectral Approximation in Sobolev Spaces
Résumé
For fixed c, Prolate Spheroidal Wave Functions $\psi_{n, c}$ form a basis with remarkable properties for the space of band-limited functions with bandwith $c$. They have been largely studied and used after the seminal work of Slepian. Recently, they have been used for the approximation of functions of the Sobolev space $H^s([-1,1])$. The choice of $c$ is then a central issue, which we address. Such functions may be seen as the restriction to $[-1,1]$ of almost time-limited and band-limited functions, for which PSWFs expansions are still well adapted. To be able to give bounds for the speed of convergence one needs uniform estimates in $n$ and $c$. To progress in this direction, we push forward the WKB method and find uniform approximation of $\psi_{n, c}$ in terms of the Bessel function $J_0$ while only point-wise asymptotic approximation was known up to now. Many uniform estimates can be deduced from this analysis. Finally, we provide the reader with numerical examples that illustrate in particular the problem of the choice of c.
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