I have been wondering whether it is possible to extend the notion of Chebyshev polynomials to non-Euclidean domains, if such a generalization is possible, or if one already exists. I am particularly interested in Riemannian 2-manifolds.

One possible generalization might come from a manifold counterpart (if it exists) of the Chebyshev differential equation:

$(1-x^2)y'' - xy' + \alpha^2y=0$ for $|x|<1$,

the solutions of which are the sought polynomials.

To my understanding, the Chebyshev polynomials are related to the Fourier basis functions by a change of variable. This may suggest the possibility to obtain the "manifold Chebyshev polynomials" by a transformation of the Fourier basis on the manifold (the eigenfunctions of the Laplace-Beltrami operator).

Another possibility might be to identify the key properties of the Chebyshev basis and phrase its computation as some optimization problem over the Stiefel manifold. For example, the Fourier basis on manifold $M$ can be computed as the set of orthogonal smooth functions $\phi_i:M\to\mathbb{R}$ with minimal Dirichlet energy $\int_M \| \nabla_M \phi_i(x)\|^2 dx$. Is there some characterizing property of the Chebyshev polynomials that might admit such an optimization procedure?

Thank you very much!