Approximation by Spherical Functions

Results on the degree of approximation of continuous or Lipschitz-continuous functions f:S^n?1 → ? on the sphere in ?ⁿ by spherical functions of degree k are given in terms of k and n, strengthening results of Newman and Shapiro. An example of (restriction of) a norm shows the result to be the best possible in k and n. Further, results for smoother functions are discussed.