Approximation of functions on the sphere. (English. Russian original) Zbl 0728.41021
Mosc. Univ. Math. Bull. 45, No. 1, 17-23 (1990); translation from Vestn. Mosk. Univ., Ser. I 1990, No. 1, 15-23 (1990).
Let f be a function defined on the unit sphere \(S_ m\) of \(R^{m+1}\) such that \(f\in M_ p(S_ m)\). Using the finite difference of order r, the author considers the modulus of smoothness of f with respect to the \(L_ p\) norm, denoted by \(\omega_ r(f,\delta)_{m,p}\). The main result of the paper is a Jackson type theorem for the approximation of f by multi-polynomials of order at most N. If \(E_ N(f)_{m,p}\) is the best approximation of f in the mentioned circumstances, then it is proved that:
\[
E_ N(f)_{m,p}\leq M_ r\omega_ r(f,1/d^{1/m})_{m,p},
\]
where \(d=(2N+m)\left( \begin{matrix} N+m\\ m\end{matrix} \right)/(N+m)\), \(M_ r\) is a positive constant and m is an odd number.
Reviewer: I.Serb (Cluj-Napoca)
MSC:
41A50 | Best approximation, Chebyshev systems |
41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |