1 - Spectral Representation of Discrete-Time Birth–Death Chains

Summary

This chapter gives an introduction to orthogonal polynomials. It also includes the concept of the Stieltjes transform and some of its properties, which will play a very important role in the spectral analysis of discrete-time birth–death chains and birth–death processes. A section on the spectral theorem for orthogonal polynomials (or Favard’s theorem) will give insights into the relationship between tridiagonal Jacobi matrices and spectral probability measures. The chapter then focuses then on the classical families of orthogonal polynomials, of both continuous and discrete variables. These families are characterized as eigenfunctions of second-order differentials or difference operators of hypergeometric type solving certain Sturm–Liouville problems. These classical families are part of the so-called Askey scheme.