On a minimal property of trigonometric interpolation at equidistant nodes
Let \(\ell_ i(x,a)\) be the trigonometric Lagrangian fundamental polynomials of degree n corresponding to the nodes \(a=(a_ 0,...,a_{2n})\). The author gives a very short proof of the following conjecture of Bitzer [\textit{G. Bitzer}, ibid. 21, 177-182 (1979; Zbl 0429.42004)]: Apart from rotations on the unit sphere, \(f(a):=\int^{2\pi}_{0}\sum^{2n}_{i=0}\ell^ 2_ i(x,a)dx\) attains its absolute minimum f(ā)\(=2\pi\) exactly for \(a=\bar a:=(2\pi k/2n+1)_{k=0,...,2n}\).
trigonometric Lagrangian fundamental polynomials
scientific article; zbMATH DE number 3867942
scientific article