Weighted Laplace Spaces for Spectral Measures and Rational Approximation

Abstract

We introduce the weighted Laplace space $H_w$, an RKHS of Laplace transforms on $(0,\infty )$, and study spectral measures in its dual space $H_w^{\prime }$. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality $\Vert {\mu }_h{\Vert }_{H_w^{\prime }}\le \Vert \mu {\Vert }_{H_w^{\prime }}$, where $\mu ={\sum }_k{\delta }_{{\lambda }_k}$ and ${\mu }_h={\sum }_k{\delta }_{{\lambda }_{k,h}}$. The proof combines min-max monotonicity of FEM eigenvalues with a heat-trace representation of the dual norm. We then analyze $H_w$-adapted rational approximation of shifted symbols $\varphi (x)=(x+{\kappa }^2{)}^{-\beta }$ and give a conditional transfer principle for estimates proved in the corresponding weighted Laplace pre-image norm. Via dual pairing, the norm inequality yields uniform bounds for finite spectral sums and related transformed observables.

29 pages, 5 figures. Generative AI has been used for this paper