Fourier / CONCEPT

CoordinatePoisson

Let n be any finite coordinate type and f a continuous complex function on real coordinate space. Assume f is integrable for coordinate Lebesgue measure, its integer translates have summable sup norms on every compact set, and the integer samples of its negative-exponent Fourier integral are summable. Then sum over integer vectors a of f(a) equals sum over integer vectors k of the integral of exp(-2 pi i sum_j k_j x_j) f(x). The unit cube is an actual fundamental domain for integer translations. Local normal convergence justifies descent to the unit torus and dominated sum-integral interchange; the torus Fourier series reconstructs the value at zero.