On recurrence coefficients for rapidly decreasing exponential weights

E. Levin; D. S. Lubinsky

doi:10.1016/j.jat.2006.06.004

On recurrence coefficients for rapidly decreasing exponential weights

E. Levin, D. S. Lubinsky

Mathematics discipline

Research output: Contribution to journal › Article › peer-review

Abstract

Let, for example,W fenced(x) = exp fenced(- exp_k fenced(1 - x²)^{- 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 W². 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
Pages (from-to)	260-281
Number of pages	22
Journal	Journal of Approximation Theory
Volume	144
Issue number	2
DOIs	https://doi.org/10.1016/j.jat.2006.06.004
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: [email protected] (E. Levin), [email protected] (D.S. Lubinsky).

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10.1016/j.jat.2006.06.004

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@article{bc8eb796f6da4e2ca85ebf53be54139d,

title = "On recurrence coefficients for rapidly decreasing exponential weights",

abstract = "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.",

author = "E. Levin and Lubinsky, {D. S.}",

note = "Funding Information: Research supported by NSF Grant DMS0400446 and US-Israel BSF Grant 2004353. ∗Corresponding author. E-mail addresses: [email protected] (E. Levin), [email protected] (D.S. Lubinsky).",

year = "2007",

month = feb,

doi = "10.1016/j.jat.2006.06.004",

language = "אנגלית",

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AU - Levin, E.

AU - Lubinsky, D. S.

N1 - Funding Information: Research supported by NSF Grant DMS0400446 and US-Israel BSF Grant 2004353. ∗Corresponding author. E-mail addresses: [email protected] (E. Levin), [email protected] (D.S. Lubinsky).

PY - 2007/2

Y1 - 2007/2

N2 - 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.

AB - 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.

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On recurrence coefficients for rapidly decreasing exponential weights

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