Convergence properties of spline-like cardinal interpolation operators acting on $l^p$ data

If $f\in \{f\in L^p(\mathbb{R}): f(x)=\int_{-\pi}^{\pi}e^{ix\xi}d\beta(\xi), \beta\in B.V.([-\pi,\pi]) \}$, then $f$ is determined by its samples on the integers by taking an appropriate limit. Specifically, $\| f - L_{\phi_\alpha}f \|_{L^p(\mathbb{R})}\to 0$ as $\alpha\to\infty$ provided that $\{\phi_\alpha: \alpha\in A\}$ is what we call a spline-like family of cardinal interpolators.