Every even Galerkin vector induces a band-limited Guinand–Weil test function whose zero sum equals the truncated Weil form exactly, and the omitted archimedean tail is a totally positive Cauchy–Stieltjes increment with budget ~ log T / T.
On hermitian forms attached to zeta functions
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A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form
Every even Galerkin vector induces a band-limited Guinand–Weil test function whose zero sum equals the truncated Weil form exactly, and the omitted archimedean tail is a totally positive Cauchy–Stieltjes increment with budget ~ log T / T.