The geometry of nodal sets and outlier detection (Q1686159)

The authors consider a few eigenvalue problems for the Laplace operator in cases where the spectrum consists of a sequence of eigenvalues \(\{\lambda_k\}\) with eigenfunctions \(\{\phi_k\}\) such as the Dirichlet problem in the unit interval of the real line and in the unit square of the plane, and also a problem on the Paley graph over a finite field and show that the functions of the variable \(x\) in the domain of the problem of the family \[ \sum_{k\leq N}\frac{1}{\sqrt{\lambda_k}}\frac{|\phi_k(x)|}{\|\phi_k\|_\infty} \] indexed by the natural number \(N\) (or variations thereof) have minimizers at points which are shown to have some interest. The authors also discuss the use of such functions in some concrete examples such as sea mine detection and defect detection.