Example. Positive temporal weights for signs and phases [fdip-0081]
Example. Positive temporal weights for signs and phases [fdip-0081]
The transfer kernel conditions have elementary examples. For \(G=\mathbb Z_2=\{1,-1\}\) and \(\beta >0\), take \(k(u)=e^{\beta u}\). The two characters are \(1\) and \(u\), and \[k(u)=\cosh \beta +u\sinh \beta .\] Thus \(a_{\mathrm {triv}}=\cosh \beta >0\) and \(a_{\mathrm {sgn}}=\sinh \beta >0\). Both dimensions are one, so the coefficient sum is \(e^\beta \). The function is real, strictly positive and invariant under inversion.
For \(G=U(1)\), the Wilson and Villain weights give two choices: \[\begin {aligned} k_W(e^{i\theta })&=e^{\beta \cos \theta }, &a_n^W&=I_n(\beta ),\\ k_V(e^{i\theta })&=\sum _{\ell \in \mathbb Z} e^{-\beta _V(\theta -2\pi \ell )^2/2}, &a_n^V&=\frac {e^{-n^2/(2\beta _V)}}{\sqrt {2\pi \beta _V}}. \end {aligned}\] Here \(\beta ,\beta _V>0\) and \(n\in \mathbb Z\). All coefficients are positive and summable; all characters have dimension one. The cited calculation proves the series, continuity and inversion symmetry. Each function is strictly positive from its definition, so every hypothesis is met. These are unnormalized weights; no zeroth coefficient has been divided out.
At \(\beta =0\), the sign or Wilson weight is constant and its nontrivial Fourier coefficients vanish. The resulting convolution kills nonconstant modes, so strict coefficient positivity cannot be removed from the injectivity assertion in Theorem [fdip-0080].