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? Are there any other operators also provide orthogonal basis which are also useful? Are there any example that we are not using Laplace operator?

On non-periodic domain, we have many orthogonal polynomial systems, say, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials. So, we have more than just one set of orthogonal basis, in this case. It motivates me to ask those above questions.

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