Dual Bases for Analytic Bernoulli Functions

We present a dual-basis framework for analytic Bernoulli functions. On the Hurwitz side, even zeta values arise, while on the Clausen side, odd zeta values appear. Both bases are generated by the same Heisenberg–Weyl ladder and are linked by the Poisson–Lerch transform, which plays the role of a Fourier bridge. The resulting orthogonality relations isolate $\zeta (2m)$ and $\beta (2m+1)$ in strictly separated channels. Low-degree examples confirm the rational evaluations, and appendices connect the picture with selector kernels, Poisson summation, and oscillator analogies.