The Hilbert Space Asymptotics of a Class of Orthonormal Polynomials on a Bounded Interval

Abstract

We study the asymptotics of orthonormal polynomials {p _n (cos x)} ^∞_n=0 , associated with a certain class of weight functions $w\left( \mathfrak{X} \right) = 1/c\left( \mathfrak{X} \right)c\left( { - \mathfrak{X}} \right)$ on [0, π]. Our principal result is that the norm of the difference of pn(cos x) and $D_n \left( \mathfrak{X} \right) \equiv c\left( \mathfrak{X} \right)e^{in\mathfrak{X}} + c\left( { - \mathfrak{X}} \right)e^{ - in\mathfrak{X}}$ in L²([0, π], (2π)⁻¹ω(x)dx) vanishes exponentially as n → ∞. The decay rate is determined by analyticity properties of the c-function.