# On recurrence coefficients for rapidly decreasing exponential weights

E. Levin, D. S. Lubinsky

نتاج البحث: نشر في مجلةمقالةمراجعة النظراء

## ملخص

Let, for example,W fenced(x) = exp fenced(- expk fenced(1 - x2)- α), x ∈ fenced(- 1, 1),where α > 0, k ≥ 1, and expk = exp fenced(exp fenced(... exp fenced())) denotes the kth iterated exponential. Let {} fenced(An) denote the recurrence coefficients in the recurrence relationxpn fenced(x) = An pn + 1 fenced(x) + An - 1 pn - 1 fenced(x)for the orthonormal polynomials {} fenced(pn) associated with W2. We prove that as n → ∞,frac(1, 2) - An = frac(1, 4) fenced(logk n)- 1 / α fenced(1 + o fenced(1)),where logk = 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.

اللغة الأصلية الإنجليزيّة 260-281 22 Journal of Approximation Theory 144 2 https://doi.org/10.1016/j.jat.2006.06.004 نُشِر - فبراير 2007

### ملاحظة ببليوغرافية

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).

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