Consider the Sturm-Liouville problem on a finite interval with Dirichlet boundary conditions. Let the elastic modulus and the density be of bounded variation. Results for both the forward problem and the inverse problem are established. For the forward problem, new bounds are established for the eigenfrequencies. The bounds are sharp. For the inverse problem, it is shown that the elastic modulu...

We obtain novel, explicit formulas for the sensitivity of Jacobi matrices to small perturbations of their spectra. Our derivation is based on the connection between Lanczos’s algorithm and the discrete Gel’fand–Levitan inverse spectral method. We prove uniform stability of Lanczos recursions in discrete primitive norms, for perturbations of the eigenvalues relative to their separations. A stron...

We study the Al-Salam–Chihara polynomials when q > 1. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with...