A counterexample to subexponential growth of orthogonal polynomials
Two systems of orthogonal polynomials on the real line are constructed. These orthogonal polynomials satisfy a three-term recurrence relation for which most of the recurrence coefficients are constant (as for Chebyshev polynomials), except for a thin subsequence. It is shown that \([- 1, 1]\) is in the support of the orthogonality measure but that the support contains points outside \([- 1, 1]\). The author then shows that at the endpoints of the support one has \(\liminf|p_n(x)|^{1/n}> 1\). This is quite interesting since it is known that \(\limsup|p_n(x)|^{1/n}\leq 1\) uniformly for \(x\) on the support of the orthogonality measure when the recurrence coefficients are asymptotically periodic. Szwarc's examples thus show that this subexponential growth of orthogonal polynomials on the support of the measure does not hold in general for measures on a compact support and that some asymptotic control of the recurrence coefficients indeed is needed.
scientific article; zbMATH DE number 810116