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Polynomials of least deviation from zero and Chebyshev-type cubature formulas. (English. Russian original) Zbl 1001.41016

Function spaces, harmonic analysis, and differential equations. Collected papers. Dedicated to the 95th anniversary of academician S. M. Nikol’skii. Transl. from the Russian. Moscow: MAIK Nauka/Interperiodika Publishing, Proc. Steklov Inst. Math. 232, 39-51 (2001); translation from Tr. Mat. Inst. Steklova 232, 45-57 (2001).
Considering the univariate case 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.
For the entire collection see [Zbl 0981.00017].

MSC:

41A55 Approximate quadratures
41A20 Approximation by rational functions
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