Oscillatory behavior of orthogonal polynomials

Let da be a positive Borel measure in (-1,1) with a' > 0 a.e. It is shown that the polynomials p" orthonormal with respect to this measure oscillate almost everywhere in (-1,1). A function F is also described that is a pointwise bound for pn, exceeded only on sets of small measure. It is shown that F is the best possible. 1. Introduction and statement of results. Let a be a nondecreasing function on (-1,1) with infinitely many points of increase, and denote by p"(x) = p"(da, x) the polynomials orthonormal with respect to da; that is, pn is a polynomial of degree « with positive leading coefficient yn = y"(da) such that ri / PÂx)p"{x)da(x) = 8mn (m,n>0) pn(x) = A(x)sm(kn(x) + B(x)), and the oscillatory behavior of the sequence (p"(x)) can easily be deduced from this expression. However, all of the results concerning pointwise asymptotics are rather special, and the asymptotic formulas for classical polynomials and for the Pollaczek polynomials (see e.g. (14, Chapter 8 and Appendix)) indicate that there is very little hope for obtaining pointwise asymptotics without imposing some condi- tions on the measure. Nevertheless, the oscillatory behavior of the sequence { p"(x)} can be proved for a fairly general class of orthogonal polynomials. In fact, we have Theorem 1. Assume a' > 0 almost everywhere in (-1,1). Then, for almost every x G (-1,1), the set of accumulation points of the sequence {p"(x)}™=0 is an interval I(x) symmetric about the origin such that its length