Density Theorems and Nonorthogonal Expansions: Are the Same?

[Dario Ringach]

Hi!

Couldn't all those "density theorems" of networks in L^2(R) be
regarded as particular cases of nonorthogonal expansion stuff we
already know?  For instance, a representation in which we translate
and dilate a single function - a family of affine coherent states -
leads to the theory of wavelet representation; while translations and
modulations - the Weyl-Heisenberg class - leads to Gabor expansions
(see [1] for instance).

Another question: given a function phi() in a Banach space, and a
function f which we want to approximate by a linear combination of N
functions which are dilations and translations of phi().  If we call the
the linear combination g.  Does anyone know how to solve the optimization
problem of min ||f-g|| ?   I know only about heuristic approaches such as
"generalized radial basis functions"...

Thanks!
--
Dario Ringach.

[1] I. Daubechies, A. Grossman, Y. Meyer, 'Painless Nonorthogonal Expansions',
    J. Math. Phys., Vol. 27, No. 5, pp.1271-1283, May 1986.