Pretty soft question, but what are the current applications of wavelets?
I'm familiar with the typical example of signal/image compression and this seems to be what many books on the topic focus on, but I've heard a lot about a sort of resurgence in wavelet theory and am having trouble finding what exactly these new applications are.
For example, professors in my department have mentioned that wavelets have found a lot of use in sparse representations and approximation theory, and I have seen the topic of wavelet reproducing kernel Hilbert spaces mentioned on more than one occasion, but have trouble finding real information about all of this and what makes it interesting.
What are some actual results in approx theory/data/prob & stats that we've gotten using wavelets?
What are some results we've gotten in pure math using wavelets?
Links to references and direct quotes of theorems/results is much appreciated