  Timelimited functions and bandlimited functions play a fundamental role in
signal and image processing. But by the uncertainty principles, a signal cannot
be simultaneously time and bandlimited. A natural assumption is thus that a
signal is almost time and almost bandlimited. The aim of this paper is to prove
that the set of almost time and almost bandlimited signals is not excluded from
the uncertainty principles. The transforms under consideration are integral
operators with bounded kernels for which there is a Parseval Theorem. Then we
define the wavelet multipliers for this class of operators, and study their
boundedness and Schatten class properties. We show that the wavelet multiplier
is unitary equivalent to a scalar multiple of the phase space restriction
operator. Moreover we prove that a signal which is almost time and almost
bandlimited can be approximated by its projection on the span of the first
eigenfunctions of the phase space restriction operator, corresponding to the
largest eigenvalues which are close to one.
