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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On Lagrange interpolation at disturbed roots of unity


Authors: Charles K. Chui, Xie Chang Shen and Le Fan Zhong
Journal: Trans. Amer. Math. Soc. 336 (1993), 817-830
MSC: Primary 30E05; Secondary 41A05, 41A10
MathSciNet review: 1087054
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Abstract: Let $ {z_{nk}} = {e^{i{t_{nk}}}}$, $ 0 \leq {t_{n0}} < \cdots < {t_{nn}} < 2\pi $, $ f$ a function in the disc algebra $ A$, and $ {\mu _n} = \max \{ \vert{t_{nk}} - 2k\pi /(n + 1)\vert:0 \leq k \leq n\} $. Denote by $ {L_n}(f;\; \cdot )$ the polynomial of degree $ n$ that agrees with $ f$ at $ \{ {z_{nk}}:k = 0, \ldots ,n\} $ . In this paper, we prove that for every $ p$, $ 0 < p < \infty $, there exists a $ {\delta _p} > 0$, such that $ \vert\vert{L_n}(f;\cdot) - f\vert{\vert _p} = O(\omega (f;\frac{1} {n}))$ whenever $ {\mu _n} \leq {\delta _p}/(n + 1)$. It must be emphasized that $ {\delta _p}$ necessarily depends on $ p$, in the sense that there exists a family $ \{ {z_{nk}}:k = 0, \ldots ,n\} $ with $ {\mu _n} = {\delta _2}/(n + 1)$ and such that $ \vert\vert{L_n}(f;\cdot) - f\vert{\vert _2} = O(\omega (f;\frac{1} {n}))$ for all $ f \in A$, but $ \sup \{ \vert\vert{L_n}(f;\cdot)\vert{\vert _p}:f \in A,\vert\vert f\vert{\vert _\infty } = 1\} $ diverges for sufficiently large values of $ p$. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for $ \{ {z_{nk}}\} $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1087054-8
PII: S 0002-9947(1993)1087054-8
Keywords: Disturbed roots of unity, Marcinkiewicz-Zygmund type inequality, Lagrange interpolation, order of approximation, $ {A_p}$-weights, $ {H^p}$-interpolation
Article copyright: © Copyright 1993 American Mathematical Society