My question is a bit general/vague.

It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (see, e.g., the famous result of DeVore, Jawerth and Popov for Besov regularity in terms of the rate of decay for the error of the best n-term approximation).

I would like to know if there is a standard well-developed theory which allows to obtain regularity results thanks to rates of decay of approximation.

Are there any **general** references in this direction ?

Many thanks in advance.

definedin these terms, e.g., Holder (C^\alpha) functions are simply those that can be approximated in L^\infty by constant functions in every ball of radius r up to an error of O(r^\alpha). If you relax L^\infty to L^2 (in an appropriate sense) you get the space H^\alpha. The space C^{k,\alpha} is the space of functions which can, in every ball, be approximated in L^\infty by a polynomial of degree k up to an error of O(r^{k+\alpha}). $\endgroup$ – Scott Armstrong Jul 23 '20 at 19:58