Sparse representation for continuous function?

Question

I recently came across the field of "Sparse representation". A talk is given here : https://www.youtube.com/watch?v=2bW4TkfTk-M.

The goal of sparse representation is taking a signal and representing without using a full generators in the linear algebra sense, so just a small subset of the whole generators.

The talk for sake of simplicity and engineering applications is mainly framed with matrix and vectors, so finite dimensional vector spaces.

Is there however a treatment of such problem (sparsity) using functional analytic tools? I think wavelets are an example of it, but I wonder if there's something more general.

Answer 1

Sure, there is a lot about this, but maybe not under that name. You can find related ideas under the name of "best $k$-term approximation", for example in Ron DeVores Acta Numerica paper "Nonlinear Approximation". Although the paper does not treat sparsity per se, the best $k$-term approximation is exactly the notion that says how good you can approximate a continuous functions with $k$ building blocks and this will lead you down the rabbit hole of Besov spaces in the case where the building blocks are wavelets.

Also the chapter "An Approximation Tour" in Mallat's book "A wavelet tour of signal processing" (Second Edition) may be worth a read. Also, compressed sensing has also been investigated for continuous signals, e.g. with Fourier or polynomial bases.