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

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, $\|\mathcal {L}_{m}\|_{1}=\sup \{\|\mathcal {L}_{m} y\|_{L(\mathbb {R})}\big / \|y\|_{l(\mathbb {Z})}\}$, is called a Lebesgue constant from $l_{1}(\mathbb {Z})$ to $L_{1}(\mathbb {R})$, and it was proved that $\sup \limits _{m} \{\|\mathcal {L}_{m}\|_{1}\}=\infty$. It is proved in this paper that \[ \sup _{m}\big \{\|\mathcal {L}_{m} y\|_{1(\mathbb {R})}\big /(\|y\|_{l(\mathbb {Z})}+\|\{H(-1)^{j} y_{j}\}\|_{l(\mathbb {Z})})\big \} \leqslant \Big (1+\frac {\pi }{2}\Big )\Big (1+\frac {\pi }{3}\Big ). \]