Sharp Estimates for Jacobi Matrices and Chain Sequences Original Research Article

Chain sequences are positive sequences {an} of the form an=gn(1−gn−1) for a nonnegative sequence {gn}. This concept has been introduced by Wall in connection with continued fractions. These sequences are very useful in determining the support of orthogonality measure for orthogonal polynomials. Equivalently, they can be used for localizing spectra of Jacobi matrices associated with orthogonal polynomials through the recurrence relation. We derive sharp estimates for chain sequences which in turn give sharp estimates for the norms of Jacobi matrices. We also give applications to unbounded essentially self-adjoint Jacobi matrices. In particular, we show how to determine whether their spectrum admits gaps around 0, and derive some integrability properties of the spectral measure.