Weighted Laplace Spaces for Spectral Measures and Rational Approximation
We introduce the weighted Laplace space $H_w$, an RKHS of Laplace transforms on $(0,\infty)$, and study spectral measures in its dual space $H'_w$. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality $\|\mu_h\|_{H'_w} \leq \|\mu\|_{H'_w}$, 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.