## Abstract

Let, for example,W fenced(x) = exp fenced(- exp_{k} fenced(1 - x^{2})^{- α}), x ∈ fenced(- 1, 1),where α > 0, k ≥ 1, and exp_{k} = exp fenced(exp fenced(... exp fenced())) denotes the kth iterated exponential. Let {} fenced(A_{n}) denote the recurrence coefficients in the recurrence relationxp_{n} fenced(x) = A_{n} p_{n + 1} fenced(x) + A_{n - 1} p_{n - 1} fenced(x)for the orthonormal polynomials {} fenced(p_{n}) associated with W^{2}. We prove that as n → ∞,frac(1, 2) - A_{n} = frac(1, 4) fenced(log_{k} n)^{- 1 / α} fenced(1 + o fenced(1)),where log_{k} = log fenced(log fenced(... log fenced())) denotes the kth iterated logarithm. This illustrates the relationship between the rate of convergence to frac(1, 2) of the recurrence coefficients, and the rate of decay of the exponential weight at ± 1. More general non-even exponential weights on a non-symmetric interval fenced(a, b) are also considered.

Original language | English |
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Pages (from-to) | 260-281 |

Number of pages | 22 |

Journal | Journal of Approximation Theory |

Volume | 144 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2007 |

### Bibliographical note

Funding Information:Research supported by NSF Grant DMS0400446 and US-Israel BSF Grant 2004353. ∗Corresponding author. E-mail addresses: elile@openu.ac.il (E. Levin), lubinsky@math.gatech.edu (D.S. Lubinsky).