Journal of Approximation Theory
| Hi-index | 0.01 |
We study ratio asymptotics, that is, existence of the limit of Pn+1(z)/Pn(Z) (Pn = monic orthogonal polynomial) and the existence of weak limits of p2ndµ (pn = Pn/||Pn||) as n → ∞ for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z0 with Im(z0) ≠ 0 implies dµ is in a Nevai class (i.e., an → a and bn → b where an, bn, are the off-diagonal and diagonal Jacobi parameters). For µ's with bounded support, we prove p2n dµ has a weak limit if and only if lim bn, lim a2n, and lim a2n+1 all exist. In both cases, we write down the limits explicitly.