On constructing distribution functions: A bounded denumerable spectrum with<i>n</i>limit points

Abstract

If {an}™ and {bn}T are real sequences with the &%'s all positive, then a theorem of Favard states that there exists a bounded increasing function ψ(x) which is a distribution function for the polynomial set {φn}~i which is recursively defined as follows: φ-ι(x) = 0, φo(x) = 1, (1-A) φn+ί(x) = (δC — an)φn(x) — bnφn-l(%) (ft ^ 0) . This study considers the problem of constructing ψ(x) for certain classes of sequences {an}T and {bn}?. The sequences considered all lead to functions ψ(x) which have a bounded denumerable spectrum with n limit points (1 ^ n < oo).