Abstract
We discuss the geometry of Laplacian eigenfunctions on compact manifolds (M, g) and combinatorial graphs . The “dual” geometry of Laplacian eigenfunctions is well understood on (identified with ) and (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of “similarity” between eigenfunctions and is given by a global average of local correlations where is the classical heat kernel and . This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.
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