= Gaussian Dirichlet energy
{c}
{title2=$\mathcal E_\gamma(f)=\int|f'|^2\,d\gamma=\sum_{n\geq1}n|c_n|^2$}
The energy form of the <Gaussian number operator> is the displayed squared <derivative> integral. Its form domain requires $\sum n|c_n|^2<\infty$, while the full <operator domain> requires $\sum n^2|c_n|^2<\infty$. It can be defined by $\lim_{t\downarrow0}t^{-1}(\|f\|_2^2-\langle f,P_tf\rangle)$ even when $Nf$ does not exist in $L^2$. This energy occurs in the <Gaussian Poincaré inequality> and <Gaussian logarithmic Sobolev inequality>.
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