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The $p$-biased Fourier coefficient of $f$ at $S$ is ${\hat{f}}_p(S)={\mathbb{E}}_{{\mu }_p}[f{\varphi }_S]$.

Every bounded operator $S$ defines a functional $T\mapsto \mathrm{tr}(ST)$ on the trace-class operators, andFinite-rank density in ${\mathcal{S}}_1$ makes every continuous functional arise uniquely this way.

Adding the correctiongivesIts common-basis eigenvalues areSince ${\lambda }_p,{\lambda }_q<0$, one has $0<d_{pq}\le 1$ for every $\mu >0$.

On the $m$ by $m$ Dirichlet grid with $h=1/(m+1)$, the exact discrete stability condition for $m>1$ isThe mesh-independent sufficient condition is $0<\mu \le 1/2$, and the exact bound tends to $1/2$ as $h\to 0$.