Not my field either but using (Chebyshev) polynomials to approximate _exponential functionals_ on [believably] infinite or scale-free graphs sounds rather interesting..

On further reflection, as you said, wavelets are local on space and phase-space so one has a natural cutoff..

Related (stub and theory warning, but closer to my domain of interest :) https://en.wikipedia.org/wiki/Analysis_on_fractals

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.12...