We start with trigonometric waveform libraries. These are localized sine
transforms (LST) associated to covering by intervals of 
(more
generally, of a manifold).
We consider a cover 
, with
and 
, write
and let 
be a window function
supported in 
such that
and
then the functions
form an orthonormal basis of 
subordinate to the partition 
. The
collection of such bases forms a library of orthonormal bases (cf. [7]).
It is easy to check that if 
denotes the space of
functions spanned by 
then
is spanned by the functions
where
is a ``window'' function covering the interval 
.
