Cardinal spline interpolation from H¹(Z) to L₁(R)

Let H¹(Z) be the discrete Hardy space, consisting of those sequences y=ⱼ∈ Z∈ lₚ(Z), such that Hy = \ Hyⱼ\∈ l₁(Z), where Hyⱼ=∑ k≠ j (k-j)⁻¹yⱼ, j∈ Z, is the discrete Hilbert transform of y. For a sequence y=ⱼ\∈ l₁(Z), let Lₘ y(x)∈ Lₚ(R) be the unique cardinal spline of degree $m-1$ interpolating to y at the integers. The norm of this operator, \|Lₘ\|₁= \\|Lₘ y\|L(R) / \|y\|l(Z)\, is called a Lebesgue constant from l₁(Z) to L₁(R), and it was proved that ₘ \\|Lₘ\|₁\=∞. It is proved in this paper that \[ ₘ \{\|Lₘ y\|1(R) /(\|y\|l(Z)+\|\{H(-1)ʲ yⱼ\}\|l(Z)) \} (1+π /2 ) (1+π /3 ). \]