Exemplu

Un caz interesant este cel al unei convolutii gaussiene in (4.3.24):

$${\omega _{km}=}\frac 1{\Psi _0(\sigma )}\exp (-\frac{(k-m)^2}{\sigma ^2})$$ (4.3.40)

unde $\Psi _0(\sigma )=\sum\limits_{m\in \mathbb{Z}}\exp (-\frac{k^2}{\sigma ^2})$ este o functie a carui comportament asimptotic poate fi aflat prin calcul numeric direct [33,221] ( $\Psi _0(\sigma )\cong 1$ for $\sigma \in [0;0.4]$ si $\Psi _0(\sigma )\cong \sqrt{\pi }\sigma$ pentru $\sigma \in [0.8;\infty )$. Daca definim $\Psi _{1/2}(\sigma )=\sum\limits_{m\in \mathbb{Z}}\exp (-\frac{(k+\frac 12)^2}{\sigma ^2})$ obtinem $\Psi _{1/2}(\sigma )\cong 0$ pentru $\sigma \in [0;0.2]$ si $\Psi _{1/2}(\sigma )\cong \sqrt{\pi }\sigma$ pentru $\sigma \in [0.8;\infty )$). In acest caz:

$$\langle k\vert\widehat{\mathcal{F}}_d\vert m\rangle =\frac 1{\sqrt{\Psi _0(\sigma )} }\exp (-\frac{(k-m)^2}{2\sigma ^2})$$ (4.3.41)