Moduli of smoothness and approximation on the unit sphere and the unit ball (Q977500)

The authors introduce and study a new modulus of smoothness and \(K\)-functional on the sphere. Their equivalence and characterization of best approximation in terms of them are proved. Defining the new modulus of smoothness for a doubling weight by taking the norm in a weighted \(L^p\) space, it is shown that the results in [\textit{F. Dai}, Constr. Approx. 24, No. 1, 91--112 (2006; Zbl 1116.41010)] can be established using the new modulus of smoothness. A new modulus of smoothness and \(K\)-functional on the ball are deduced from the results on the sphere [cf. \textit{Z. Ditzian} and \textit{V. Totik}, Modulus of smoothness. New York etc.: Springer-Verlag (1987; Zbl 0666.41001)]. The connection between the modulus of smoothness introduced by the authors and that of Ditzian-Totik is also given. Examples of functions for which asymptotic orders of the new modulus of smoothness and best approximation by polynomials are explicitly determined are also given.