Polynomials of least deviation from zero and Chebyshev-type cubature formulas (Q2783066)

Considering the univariate case \textit{P. L. Chebyshev} [Polnoe sobranie sochinenii (Complete Works), Moscow; Leningrad: Akad. Nauk SSSR (1948)] determined the deviation of \(x^n\) from the space of algebraic polynomials of lower degrees in the space \( C[-1,1] \). The authors first give another proof of the foregoing result which is useful in the context of the multivariate case. Thus the problem about a polynomial of least deviation from zero on a multidimensional sphere is solved in some special cases. Considering some known spherical configurations the authors determine the entire family of subspaces \(H_q\), \(q \in\mathbb{N} \), of homogeneous harmonic polynomials of degree \(q\) for which the Chebyshev-type cubature formulas on the sphere are exact.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00017].