Example. A positive weight with a negative coefficient [fdip-007J]

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On \(U(1)\), take \(0<\epsilon <1\) and \[k(e^{i\theta })=1-\epsilon \cos \theta =1-\frac \epsilon 2 e^{i\theta }-\frac \epsilon 2 e^{-i\theta }.\] This is a real continuous central function, unchanged by inversion, and \(k\ge 1-\epsilon >0\). Its character coefficients at \(n=\pm 1\) are nevertheless \(-\epsilon /2\). Thus the pointwise and Fourier positivity assumptions in the kernel definition are distinct.