PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 10, October 2006, Pages 2931–2941 S 0002-9939(06)08421-8 Article electronically published on April 11, 2006 ON GENERALIZED HYPERINTERPOLATION ON THE SPHERE FENG DAI (Communicated by Andreas Seeger) Abstract. It is shown that second-order results can be attained by the gen- eralized hyperinterpolation operators on the sphere, which gives an affirmative answer to a question raised by Reimer in Constr. Approx. 18(2002), no. 2, 183–203. 1. Introduction Let S d−1 = {x ∈ R d : |x| =1} (d ≥ 3) be the unit sphere of R d endowed with the usual rotation invariant measure dσ(x). We denote by H d k the space of all spherical harmonics of degree k on S d−1 and Π d N the space of all spherical polynomials of degree at most N . The spaces H d k are mutually orthogonal with respect to the inner product 〈f,g〉 =  S d-1 f (x)g(x) dσ(x), and the space Π d N can be expressed as a direct sum (1.1) Π d N = H d 0 ⊕H d 1 ⊕···⊕H d N . Also, it is known that for f ∈H d k , f (x)=  S d-1 f (y)G k (x · y) dσ(y), x ∈ S d−1 , where here and throughout, G k (t)= 2k + d − 2 (d − 2)|S d−1 | P d-2 2 k (t), t ∈ [−1, 1], |S d−1 | is the surface area of S d−1 , and P d-2 2 k (t) is the usual ultraspherical polynomial of degree k and index d−2 2 , as defined in [Sz, p. 80]. Thus, by (1.1), we have, for f ∈ Π d N , (1.2) f (x)=  S d-1 f (y)E N (x · y) dσ(y), x ∈ S d−1 , Received by the editors April 23, 2005. 2000 Mathematics Subject Classification. Primary 41A15, 41A17; Secondary 41A05, 46E22. Key words and phrases. Spherical polynomials, generalized hyperinterpolation, second-order moduli of smoothness, unit sphere. The author was supported in part by the NSERC Canada under grant G121211001. c 2006 American Mathematical Society Reverts to public domain 28 years from publication 2931 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use