Several kinds of Chebyshev polynomials in higher dimensions

Chebyshev polynomials are arguably the most useful orthogonal polynomials for computational purposes. In one dimension they arise from the close relationship that exists between Fourier series and polynomials. We describe how this relationship generalizes to Fourier series on certain symmetric lattices, that exist in all dimensions. The associated polynomials can not be seen as tensor-product generalizations of the one-dimensional case. Yet, they still enjoy excellent properties for interpolation, integration, and spectral approximation in general, with fast FFT-based algorithms, on a variety of domains. The first interesting case is the equilateral triangle in two dimensions (almost). We further describe the generalization of Chebyshev polynomials of the second kind, and many new kinds are found when the theory is completed. Connections are made to Laplacian eigenfunctions, representation theory of finite groups, and the Gibbs phenomenon in higher dimensions.