On Lagrange interpolation at disturbed roots of unity

Chui, Charles K.; Shen, Xie Chang; Zhong, Le Fan

doi:10.1090/S0002-9947-1993-1087054-8

On Lagrange interpolation at disturbed roots of unity
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by Charles K. Chui, Xie Chang Shen and Le Fan Zhong PDF

Trans. Amer. Math. Soc. 336 (1993), 817-830 Request permission

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 ${\mu _n} = \max \{ |{t_{nk}} - 2k\pi /(n + 1)|:0 \leq k \leq n\}$. 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 $||{L_n}(f;\cdot ) - f|{|_p} = O(\omega (f;\frac {1} {n}))$ 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 $||{L_n}(f;\cdot ) - f|{|_2} = O(\omega (f;\frac {1} {n}))$ for all $f \in A$, but $\sup \{ ||{L_n}(f;\cdot )|{|_p}:f \in A,||f|{|_\infty } = 1\}$ 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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