We will now measure the distance or good fit between a basis and a function in terms of the Shannon entropy of the expansion.
Let H be a Hilbert space.
Let 
, 
and let 
be an orthogonal
decomposition of H. We define
the entropy of v relative to the decomposition 
of H, as a
measure of the distance between v and the orthogonal decomposition.
is characterized by the Shannon equation, which is a
version of Pythagoras' theorem. Let
i.e. 
and 
give orthogonal decompositions 
, 
.
Then
This is Shannon's equation for entropy (if we interpret as
in quantum
mechanics 
as the ``probability''
of v to
be in the subspace 
).
This equation enables us to search for a smallest entropy expansion of a signal.
Figure A.3: Tree search In the LST library
For example in the LST Library case, we compare the entropy of the expansion in two adjacent windows to the entropy of the expansion on their union and pick the least expensive, continuing the comparison with the selection made for the next pair, etc. (cf. figure A.3).