The chapter explores a connection between cubature rules and the discrete Fourier transform of exponential functions defined via lattice translation. Such discrete Fourier analysis yields cubature rules for exponential functions for the integral over the spectral set of the lattice, which become cubature rules on the fundamental domain of the spectral set for generalized cosine and sine functions, defined as certain symmetric and antisymmetric exponential sums. Furthermore, under appropriate transformation, these generalized trigonometric functions define Chebyshev polynomials that inherit the orthogonality of generalized sine and cosine functions, which lead to cubature rules for algebraic polynomials on the range of the fundamental domain under the transformation.

Gauss cubature rules are straightforward extensions of the Gauss quadrature rules of one variable. A Gauss cubature rule of degree 2n-1 exists if, and only if, its nodes are common zeros of all orthogonal polynomials of degree n. They are the first example of minimal cubature rules but rarely exist. The chapter gives a comprehensive study that provides a complete characterization of the Gauss cubature rules in terms of the common zeros of the orthogonal polynomials and their structural relations, and it includes examples and counterexamples for the existence of Gauss cubature rules.