We show how to improve the properties of a Hilbert space frame by projecting it onto a subspace of the Hilbert space. For example, for any frame on a n-dimensional Hilbert space, there is an orthogonal projection onto a subspace of dimension n/2 (if n is even) or (n+1)/2 (if n is odd) so that the projection of the frame becomes a tight frame.
We give a physical interpretation for finite tight frames along the lines of Columb's Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an <i>N</i>-dimensional Hilbert space which are the closest to being tight (in the sense of minimizing potential energy) while having the norms of the frame vectors prescribed in advance. This also yields a <i>fundamental inequality </i>that all finite tight frames must satisfy.
Pyramidal structures are defined which are locally a combination of low and highpass filtering. The structures are analogous to but different from wavelet packet structures. In particular, new frequency decompositions are obtained; and these decompositions can be parameterized to establish a correspondence with a large class of Cantor sets. Further correspondences are then established to relate such frequency decompositions with more general self- similarities. The role of the filters in defining these pyramidal structures gives rise to signal reconstruction algorithms, and these, in turn, are used in the analysis of speech data.