# Nonlinear discrete wavelet approximation of piecewise $C_\infty$ functions and more

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):

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 $[0,1]^2$)?
• To question 2: I am not sure about papers or explicit references, but have you tried to take a look at papers dealing with functional time series (they usually work with integral operators and analyse the spaces $C[0,1]$ and $L^2[0,1]$). Maybe one of the following authors worked on the topic you're interested in: Piotr Kokoszka, Alexander Aue, Lajos Horváth, Alexander Meister, Moritz Jirak, Denis Bosq (the book "Linear Process in Function Spaces" (2000) and "Inference and Prediction in Large Dimensions" (2007) with Delphin Blabke could be useful). Feb 11, 2017 at 20:58
• No, I was not aware of these references, I thank you for those Feb 11, 2017 at 21:04