Weighted Laplace Spaces for Spectral Measures and Rational Approximation

We introduce the weighted Laplace space $H_w$, an RKHS of Laplace transforms on $(0,\mathrm{\infty })$, and study spectral measures in its dual space $H_w^{\mathrm{\prime }}$. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality $\mathrm{\parallel }{\mu }_h{\mathrm{\parallel }}_{H_w^{\mathrm{\prime }}}\le \mathrm{\parallel }\mu {\mathrm{\parallel }}_{H_w^{\mathrm{\prime }}}$, 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 $\phi (x)=(x+{\kappa }^2{)}^{-\beta }$ 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.