BUPUs stands for bounded uniform partitions of unity.
It is the purpose of this talk to point out the importance
that come from the existence of BUPUs of arbitrary (small)
size. On the Euclidean space they can be obtained by
compressing (via dilation) any given BUPU, e.g. a partition
of unity generated by shifted (say cubic) B-splines.

They can be used to proof compactness results,
to define Wiener amalgam spaces or modulation spaces, to
understand the behavior of Shannon’s sampling theorem, or
to show how functions or measures can be approximated
by discrete measures (or discrete- periodic signals).