  Recently, there is a growing interest in the spectral approximation by the
Prolate Spheroidal Wave Functions (PSWFs) $\psi_{n, c},\, c>0.$ This is due to
the promising new contributions of these functions in various classical as well
as emerging applications from Signal Processing, Geophysics, Numerical
Analysis, etc. The PSWFs form a basis with remarkable properties not only for
the space of band-limited functions with bandwidth $c,$ but also for the
Sobolev space $H^s([-1,1])$. The quality of the spectral approximation and the
choice of the parameter $c$ when approximating a function in $H^s([-1,1])$ by
its truncated PSWFs series expansion, are the main issues. By considering a
function $f\in H^s([-1,1])$ as the restriction to $[-1,1]$ of an almost
time-limited and band-limited function, we try to give satisfactory answers to
these two issues. Also, we illustrate the different results of this work by
some numerical examples.
