real $\alpha$.

\[item:Wreg\] $\phi$ and $\psi$ are compactly-supported, integrable, $\int_{-\infty}^\infty
  \phi(t)\,\rmd t = 1$ and $\int_{-\infty}^\infty \psi^2(t)\,\rmd t = 1$.

\[item:psiHat\] There exists $\alpha>1$ such that $\sup_{\xi\in\R}|\hat{\psi}(\xi)|\,(1+|\xi|)^{\alpha} <\infty$, where $\hat{\psi}(\xi)=\int_{-\infty}^\infty\psi(t)\,\rme^{-\rmi\xi t}\rmd t$ denotes the Fourier transform of $\psi$.

\[item:MVM\] The function $\psi$ has $M$ vanishing moments, $ \int_{-\infty}^\infty t^m \psi(t) \,\rmd t=0$ for all $m=0,\dots,M-1$

\[item:MIM\] The function $ \sum_{k\in\Z} k^m\phi(\cdot-k)$ is a polynomial of degree $m$ for all $m=0,\dots,M-1$.

Under \[item:MVM\] and \[item:MIM\], the filter can be interpreted as the convolution of the $\diffop^M$ filter with a finite impulse response filter. Hence if $M\geq p$, Equation (\[eq:wavcoeffLocStat\]) may be written as $$W_{j,k;T}= \sum_{t=1}^T \tilde{h}_{j,2^jk-t}(\diffop^p X)_{t,T},\quad k=0,\dots,T_j-1\;,$$ where $h_{j,\cdot}=\tilde{h}_{j,\cdot}\ast\diffop^p$. In particular, we have $$\label{eq:HjTilde}
  \tilde{H}_j(\lambda)=\sum_{t\in\Z} \tilde{h}_{j,t}\rme^{-\rmi t\lambda} =
H_j(\lambda)(1-\rme^{\rmi\lambda})^{-p}\;.$$

Local wavelet spectrum, local scalogram, tangent scalogram, and final estimator
-------------------------------------------------------------------------------

Recall that $f(u,\cdot)$ in (\[eq:genLocSpecDens\]) can be interpreted as a *local generalized spectral density* at rescaled time $u\in[0,1]$. Hence, as in the stationary setting used in [@moulines:roueff:taqqu:2007:jtsa], for each such $u$, we may define a *local wavelet spectrum* $\sigma^2(u)=\{\sigma^2_j(u),\,j\geq0\}$, where for each $j\geq0$, $\sigma^2_j(u)$ is the variance of the wavelet coefficients at scale index $j$ of a process with generalized spectral density $f(u,\cdot)$. This variance is well defined under the 