## On a minimal property of cardinal and periodic Lagrange splines.(English)Zbl 0778.41003

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 M. Reimer [ibid. 36, 91-98 (1982; Zbl 0492.41018)] [look also D. Siepman and B. Sündermann, ibid. 39, 236-240 (1983; Zbl 0521.41007)] and I. J. Schoenberg [Linear Operators Approximation, Proc. Conf. Oberwolfach 1971, ISNM 20, 382-404 (1972; Zbl 0269.41002)].

### MSC:

 41A15 Spline approximation

### Keywords:

bounded cardinal splines; Lagrange splines

### Citations:

Zbl 0492.41018; Zbl 0521.41007; Zbl 0269.41002
Full Text:

### References:

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