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Cardinal spline interpolation from

Authors:	Fang Gensun

Institution:	Department of Mathematics, Beijing Normal University, Beijing, 100875, People's Republic of China

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

Keywords:	Cardinal spline entire function Lebesgue constant

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