Trigonometric Selector Kernels, Duality, and Odd Zeta Values

Abstract

In this short note, we develop trigonometric selector kernels to represent odd zeta values via dual hyperbolic counterparts. This framework highlights a Fourier-Poisson duality, incorporating finite-part integrals in the sense of Hadamard-Galapon. In particular, we show how such kernels naturally recover Euler-Maclaurin and Poisson summation formulas as dual manifestations. We further connect our kernel approach with the finite-part integral formulation, extending earlier Cvijovi\'c-Klinowski type representations for odd zeta values.