Approximation of functions on the sphere (Q804856)

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.