a non-negative symmetric function and $\rho_{j,T}$ is a normalizing term so that (\[eq:normWeights\]) holds. In the last display we see that $\band_T$ is the bandwidth in a rescaled time sense while $\band_{T}T_j$ is the corresponding bandwidth in the sense of location indices $k=0,1,2,\dots,T_j$ at scale $j$ . Lemma \[lem:two-sided-kernel\] in the appendix shows that Assumption \[assump:weights\] holds for a wide variety of choices for $K$. In particular Assumption \[assump:weights\] holds with $\delta_{j,T}\asymp(b_TT_j)^{-1}$ and $V(i,v,;i',v')=2\pi2^{-i-i'}$ under the following assumption.

\[assump:rect\_weights\] The weights $(\gamma_{j,T}(k))$ are defined by (\[eq:KernelWeights\]) with $K=\1_{[-1/2,1/2]}$. Moreover, as $T\to\infty$, $b_T\to0$ and $T_jb_T\to\infty$.

\[ex:recursive-kernel\] By *recursive weights*, we here mean that, given $T,\;L$ and $\bw$, the possibility of computing $\widehat{\sigma}^2_{j,T}(u)$ by successive simple linear processing involving a finite number of operations after each new observations $X_{t;T}$ as $t$ grows from $t=1$ to $t=T$.

Because the DWT is defined as a succession of finite filtering and decimation, it is indeed possible to compute $W_{j,k;T}$ online for all $j\in\{L,\dots,L+\ell\}$ and $k\in\{0, \dots,T_j\}$. Then an online implementation of the local *recursive* scalogram can be done by setting $$\widehat{\sigma}^2_{j,-1;T}=0\;, \quad j\in\{L,\dots,L+\ell\},$$ and, using the following recursive equation for all $j\in\{L,\dots,L+\ell\}$ and $t\in\{0,\dots,T_j-1\}$, $$\widehat{\sigma}^2_{j,t;T}=\exp(-(\band_{T}T_j)^{-1}) \; \widehat{\sigma}^2_{j,t-1;T} + W_{j,t;T}^2\;,$$ where $(\band_{T}T_j)^{-1}$ is the exponential *forgetting 