Take a function $f$ defined on $[0,1]$ with a finite number of discontinuities and uniformly Lipschitz $\alpha>1/2$ between discontinuities. We consider its $M$-term approximation $f_M$, in a suitable discrete orthogonal basis, using the $M$ largest (in amplitude) projection coefficients.
Essentially (I am skipping some details to references):
$$\|f-f_M \|^2 = O\left(M^{-2\alpha}\right)\,.$$
The more regular $f$, the "easiest" to approximate. Such results can be found in (I apologize for "signal processing" references):
- S. Mallat, A wavelet tour of signal processing (the sparse way), Chapter 9
- P. L. Dragotti, Wavelet footprints: theory, algorithms, and applications, 2003
- M. Vetterli, Wavelets, approximation, and compression, 2002
There are many other related results, with different types of "regularity" (bounded variation, Sobolev) and extensions to higher dimensions.
My questions are:
- What are the approximation rates for piecewise 1D $C_\infty$ functions (with references)?
- Bonus: is there a compendium or survey of such approximation results in orthonormal and wavelet bases for piecewise (with a finite set of pieces) functions in $[0,1]$ (and possibly $[0,1]^2$)?