VON NEUMANN INDICES AND CLASSES OF POSITIVE DEFINITE FUNCTIONS
Palle E. T. Jørgensen, Feng Tian · 2016
With view to applications, we establish a correspondence between two problems: (i) the problem of finding continuous positive definite extensions of functions F which are defined on open bounded domains Ω in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}$\end{document}R, on the one hand; and (ii) spectral theory for elliptic differential operators acting on Ω (constant coefficients). A novelty in our approach is the use of a reproducing kernel Hilbert space \documentclass[12pt]{minimal}\begin{document}$\mathscr {H}_{F}$\end{document}HF computed directly from (Ω, F), as well as algorithms for computing relevant orthonormal bases in \documentclass[12pt]{minimal}\begin{document}$\mathscr {H}_{F}$\end{document}HF.