The asymptotic connection between approximation by cardinal splines and entire functions of exponential type (Q1913926)

Denote by \(L_p(\mathbb{R})\), \(1\leq p\leq \infty\) the classical Lebesgue spaces endowed with the norms \(|\cdot|_{p(\mathbb{R})}\). For \(m\in \mathbb{N}\), \(h>0\) denote by \(S_{h,m}\) the set of all cardinal splines of degree \(m\) with nodes \(\{(j+m/2)h\}_{j\in \mathbb{Z}}\). For \(\sigma>0\) denote by \(\varepsilon_\sigma(\mathbb{R})\) the restriction on \(\mathbb{R}\) of entire functions of exponential type \(\sigma\) and let \(B_{\sigma,p}:=\varepsilon_\sigma(\mathbb{R})\cap L_p(\mathbb{R})\), \(1\leq p\leq\infty\). Also if \(F\) is a subset of a normed linear space \(X\) and \(f\in X\), denote by \(E(f,F)_X\) the best approximation of \(f\) by elements of \(F\). The main results are the following: Theorem 1. There is a function \(f_0\in C(\mathbb{R})\cap L_\infty(\mathbb{R})\) such that \[ \lim_{m\to\infty} E(f_0,S_{\pi/\sigma,m})_{\infty(\mathbb{R})}>E(f_0,B_{\sigma,\infty})_{\infty(\mathbb{R}) }, \] (here \(f_0=\sin\)) and Theorem 2. If \(f\in L_p(\mathbb{R})\), \(1<p<\infty\), then \[ \lim_{m\to\infty} E(f,S_{\pi/\sigma,m}\cap L_p(\mathbb{R}))_{p(\mathbb{R})}\leq E(f,B_{\sigma,p})_{p(\mathbb{R})};\tag{1} \] (2) In the case \(p=2\) there is equality in (1).