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This article is cited in 10 scientific papers (total in 10 papers)
The Taikov Functional in the Space of Algebraic Polynomials on the Multidimensional Euclidean Sphere
M. V. Deikalova Ural State University
Abstract:
We discuss three related extremal problems on the set $\mathscr P_{n,m}$ of algebraic polynomials of given degree $n$ on the unit sphere $\mathbb S^{m-1}$ of Euclidean space $\mathbb R^m$ of dimension $m\ge 2$. (1) The norm of the functional $F(h)=F_hP_n=\int_{\mathbb C(h)}P_n(x)\,dx$, which is equal to the integral over the spherical cap $\mathbb C(h)$ of angular radius $\operatorname{arccos} h$, $-1<h<1$, on the set $\mathscr P_{n,m}$ with the norm of the space $L(\mathbb S^{m-1})$ of summable functions on the sphere. (2) The best approximation in $L_\infty(\mathbb S^{m-1})$ of the characteristic function $\chi_h$ of the cap $\mathbb C(h)$ by the subspace $\mathscr P^\bot_{n,m}$ of functions from $L_\infty(\mathbb S^{m-1})$ that are orthogonal to the space of polynomials $\mathscr P_{n,m}$. (3) The best approximation in the space $L(\mathbb S^{m-1})$ of the function $\chi_h$ by the space of polynomials $\mathscr P_{n,m}$. We present the solution of all three problems for the value $h=t(n,m)$ which is the largest root of the polynomial in a single variable of degree $n+1$ least deviating from zero in the space $L_1^\phi$ on the interval $(-1,1)$ with ultraspheric weight $\phi(t)=(1-t^2)^\alpha$, $\alpha=(m-3)/2$.
Keywords:
Taikov functional, algebraic polynomial, Euclidean sphere, spherical cap, polynomial of least deviation, Lebesgue measure, Hahn–Banach theorem, zonal function.
Received: 31.12.2007 Revised: 11.01.2008
Citation:
M. V. Deikalova, “The Taikov Functional in the Space of Algebraic Polynomials on the Multidimensional Euclidean Sphere”, Mat. Zametki, 84:4 (2008), 532–551; Math. Notes, 84:4 (2008), 498–514
Linking options:
https://www.mathnet.ru/eng/mzm6137https://doi.org/10.4213/mzm6137 https://www.mathnet.ru/eng/mzm/v84/i4/p532
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