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For fixed $c,$ the Prolate Spheroidal Wave Functions (PSWFs) $\psi_{n, c}$form a basis with remarkable properties for the space of band-limited functionswith bandwidth $c$. They have been largely studied and used after the seminalwork of D. Slepian, H. Landau and H. Pollack. Recently, they have been used forthe approximation of functions in the Sobolev space $H^s([-1,1])$. In view ofthis, we give new estimates on the decay rate of eigenvalues of the Sinc kernelintegral operators. This is one of the main issues of this work. A second oneis the choice of the parameter $c$ when approximating a function in$H^s([-1,1])$ by its truncated PSWFs series expansion. Such functions may beseen as the restriction to $[-1,1]$ of almost time-limited and band-limitedfunctions, for which PSWFs expansions are still well adapted. Finally, weprovide the reader with some numerical examples that illustrate the differentresults of this work.

Spectral decay of the sinc kernel operator and approximations by Prolate Spheroidal Wave Functions