Expanding piecewise polynomial using Daubechies wavelet What is the best Daubechies wavelet (i.e. the number of vanishing moment) to expand a signal $\boldsymbol{x} \in \mathbb{R}^n$? 
$\boldsymbol{x}$ consists of $m$ pieces of polynomial with $d$ degree. The criterium is to make the DWT signal as sparse as possible.

Update:
The goal of sparsifying the signal in wavelet domain is denoising. Let $\boldsymbol{W}$ denote DWT. 
$$
\boldsymbol{y} = \boldsymbol{Wx}
$$
Apply a soft-thresholding to $\boldsymbol{y}$,
$$
\hat{\boldsymbol{y}} = \text{sign}(\boldsymbol{y})(\max(\boldsymbol{y}-\lambda,0))
$$
Choose $\lambda=\sqrt{2\log n}$ according to Ideal spatial adaptation by wavelet shrinkage.
The sparsity is defined by $\| \hat{\boldsymbol{y}} \|_0$.
According to your formula, you also apply soft-thresholding to the approximation coefficients, which is not standard. Aside, your operator $W$ does not seem to specify the number of wavelet levels used. Finally, your class of signals does not seem to address the regularity at piecewise junctions. 

I believe in this case very unlikely that in a discrete implementation, without further relation between $m$ and $n$, you can find, theoretically,  a best wavelet in all cases, because $\|\hat{y}\|_0$ is a quite sensitive index (and it is not a norm).  Of course, a Daubechies with $d$ vanishing moments would seem appropriate.

But since the DWT is quite fast, in your context, you could just find a "generally best wavelet",  by simulating random signals, and iterating over some levels, and each Daubechies wavelet with moments in $[d-2,\ldots,d+2]$ for instance.