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

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