Stability estimates for the Jacobi inverse eigenvalue problem

We present different stability estimates for the Jacobi inverse eigenvalue problem. First, we give upper bounds expressed in terms of quadrature data and not having weights in denominators. The technique of orthonormal polynomials and integral representation of Hankel determinants is used. Our bounds exhibit only polynomial growth in the problem’s dimension (see [4]). It has been shown that the approach, exploiting integral representation of Hankel determinants, implies at least quantitative improvement of earlier results and generalizes them (see [5]). It has also been demonstrated that a particular implementation of the Hankel determinant approach gives an estimate being unimprovable up to a coefficient; the corresponding example involves quadrature data with a small but not too small weight. It follows that polynomial increase of a general case upper bound in terms of the dimension is unavoidable (see [5]). Second, we derive a differential formula for perturbations which allows us to effectively handle the special case of a spectral measure with approximately equal weights (see [2]). This special case is closely related to the inverse discrete SturmLiouville problem (see [1]), considered in combination with the ”optimal finite difference grids” technique [3].