Published online by Cambridge University Press: 10 October 2025
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.
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