On a minimal property of cardinal and periodic Lagrange splines (Q1198159)

Let \(S_ n\) be the space of bounded cardinal splines and \(S^{(N)}:=\sum^ N_ 1\bigl(L_ j^{(N)}(x)\bigr)^ 2\), where \(L_ j^{(N)}:=L^{(N)}(\cdot-j)\) is the \(N\)-periodic cardinal spline satisfying \(L^{(N)}_ j(k)=\delta_{kj}\), \(k=1,\ldots,N\). It is proved that \(S^{(N)}=\sum^ N_{j=1}| Sf_ j|^ 2/N\), where \(Sf_ j\) is a cardinal spline interpolant at the integers to the complex exponential \(f_ j(x):=\exp(2\pi ijx/N)\). This allows to establish that Lagrange splines \(L_ j:=L(\cdot-j)\) form extremal basis for the spaces \(S_ N\) of given degree \(n\) in the sense that \(\| L\|_ \infty=1\). The last statement was earlier established by \textit{M. Reimer} [ibid. 36, 91-98 (1982; Zbl 0492.41018)] [look also \textit{D. Siepman} and \textit{B. Sündermann}, ibid. 39, 236-240 (1983; Zbl 0521.41007)] and \textit{I. J. Schoenberg} [Linear Operators Approximation, Proc. Conf. Oberwolfach 1971, ISNM 20, 382-404 (1972; Zbl 0269.41002)].