  For fixed $c,$ the Prolate Spheroidal Wave Functions (PSWFs) $\psi_{n, c}$
form a basis with remarkable properties for the space of band-limited functions
with bandwidth $c$. They have been largely studied and used after the seminal
work of D. Slepian, H. Landau and H. Pollack. Recently, they have been used for
the approximation of functions in the Sobolev space $H^s([-1,1])$. In view of
this, we give new estimates on the decay rate of eigenvalues of the Sinc kernel
integral operators. This is one of the main issues of this work. A second one
is the choice of the parameter $c$ when approximating a function in
$H^s([-1,1])$ by its truncated PSWFs series expansion. 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. Finally, we
provide the reader with some numerical examples that illustrate the different
results of this work.
