Some remarks on the Fejér problem for Lagrange interpolation in several variables (Q919554)

Let \(P_ n(X)\), \(X\subset {\mathbb{R}}^ n\) compact, denote the vector space of the polynomials of total degree at most n, when restricted to X. Suppose that \(N(=N_ m(X))\) is the dimension of \(P_ n(X)\). If \(\{p_ 1,...,p_ N\}\) is a basis of \(P_ n(X)\) and \(\{x_ 1,...,x_ N\}\) a collection of N points of X, then the \(N\times N\) matrix \(V_ n(x_ 1,...,x_ N):=(p_ i(x_ j))\) is known as the corresponding Vandermonde matrix and \(VDM(x_ 1,...,x_ N):=\det V_ n(x_ 1,...,x_ n)\) is the Vandermonde determinant. If \(VDM(x_ 1,...,x_ N)\neq 0\), we may form the Lagrange polynomials \(\ell_ i\). \(\Lambda (x):=\sum^{N}_{i=1}| \ell_ i(x)|\) is known as the Lebesgue function of interpolation. \textit{L. Fejer} [Ann. Sc. Norm. Super. Pisa, II. Ser. 1, 263-276 (1932; Zbl 0004.24903)] proved the remarkable fact that, for the unit circle X, at the points which maximize the Vandermonde determinant, \(\max_{x\in X}| \ell_ i(x)| =1\) and hence \(\max_{x\in X}\Lambda (x)\leq \sqrt{N}\). The note is concerned with the question if \(\max_{x\in X}\sum^{N}_{i=1}\ell^ 2_ i(x)=1\) for other regions. In particular, the unit sphere and the unit ball are considered. The answer to these questions is related to the theory of optimal experimental design and to the theory of tight spherical design [see, for example, \textit{S. Karlin}, \textit{W. J. Studden}, Tschebyscheff Systems: With applications in analysis and statistics. (1966; Zbl 0153.389) and \textit{P. Delsarte}, \textit{J. M. Goethals}, \textit{J. J. Seidel}, Geom. Dedicata 6, 363-388 (1977; Zbl 0376.05015)]. The purpose of the note is to collect the results relevant to interpolation and use them to conclude that, except in exceptional circumstance, \(\max_{x\in X}\sum^{N}_{i=1}\ell^ 2_ i(x)>1\) for the aforementioned two cases.