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Cardinal spline interpolation from $H^{1}(\mathbb{Z} )$ to $L_{1}(\mathbb{R} )$

Author: Fang Gensun
Journal: Proc. Amer. Math. Soc. 128 (2000), 2597-2601
MSC (2000): Primary 41A17, 42B30; Secondary 30D15, 30D55
Published electronically: February 21, 2000
MathSciNet review: 1657739
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $H^{1}(\mathbb{Z} )$ be the discrete Hardy space, consisting of those sequences $y=\{y_{j}\}_{j\in \mathbb{Z} }\in l_{p}(\mathbb{Z} )$, such that $Hy = \{ Hy_{j}\}\in l_{1}(\mathbb{Z} )$, where $Hy_{j}=\sum \limits _{k\ne j} (k-j)^{-1}y_{j}$, $j\in \mathbb{Z} $, is the discrete Hilbert transform of $y$. For a sequence $y=\{y_{j}\}\in l_{1}(\mathbb{Z} )$, let $\mathcal{L}_{m} y(x)\in L_{p}(\mathbb{R} )$ be the unique cardinal spline of degree $m-1$interpolating to $y$ at the integers. The norm of this operator, $\Vert\mathcal{L}_{m}\Vert _{1}=\sup \{\Vert\mathcal{L}_{m} y\Vert _{L(\mathbb{R} )}\big / \Vert y\Vert _{l(\mathbb{Z} )}\}$, is called a Lebesgue constant from $l_{1}(\mathbb{Z} )$ to $L_{1}(\mathbb{R} )$, and it was proved that $\sup \limits _{m}\,\{\Vert\mathcal{L}_{m}\Vert _{1}\}=\infty $.

It is proved in this paper that

\begin{displaymath}\sup _{m}\big \{\Vert\mathcal{L}_{m} y\Vert _{1(\mathbb{R} )... ...lant \Big (1+\frac{\pi }{2}\Big )\Big (1+\frac{\pi }{3}\Big ). \end{displaymath}

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Additional Information

Fang Gensun
Affiliation: Department of Mathematics, Beijing Normal University, Beijing, 100875, People’s Republic of China

Keywords: Cardinal spline, entire function, Lebesgue constant
Received by editor(s): January 21, 1997
Received by editor(s) in revised form: October 13, 1998
Published electronically: February 21, 2000
Additional Notes: Project 19671012 supported by both the National Natural Science Foundation and the Doctoral Programme Foundation of Institution of Higher Education of the People’s Republic of China
Communicated by: J. Marshall Ash
Article copyright: © Copyright 2000 American Mathematical Society

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