Consider the Hilbert space associated with the $L^p$-norm. I'm interested to find the best (in the sense of the most sparse) truncated approximation for isomorphic functions in this space. Naturally, I have to pay attention to the best basis set I can choose.
I know that for a complete orthonormal basis, one can find a truncated basis set $B_T$ which approximates the space elements $v$ up to the tolerance level $T = ||Proj(v,B_T^{\perp})||_p$ (I'm misusing the notation here...)
I have two related questions:
- Do orthogonal basis always give the most sparse representation, for a fixed function and space?
- what are the best strategies to find the best basis set given some information about the functions?
For instance, for functions with the following property:
- Having most of their (p-norm) energy in the lower (or higher) frequencies
- oscillatory decaying functions
- pieces-wise constant with limited domain $D = [d_1,d_2]$
- unknown properties (Can I for instance use Hermite functions as a basis if $p=2$, for instance?)
Please don't hesitate to link or introduce resources that cover this topic. Thank you.