Oscillatory behavior of orthogonal polynomials

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