On periodic domain, people always use Fourier basis, which eigenvectors of Laplace operator. On sphere, people use spherical harmonics, which also are eigenvectors of Laplace operator. In applied science, people decompose functions on a graph using eigenvectors of graph laplacian.

What makes eigenvectors of Laplace operator widely used compared to other orthogonal basis? Do we have other choices for the basis? Does any orthogonal basis do the same job?

On non-periodic domain, we have many orthogonal polynomial systems, say, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials. So, we have choices. It motivates me to ask this question.