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On generalized hyperinterpolation on the sphere

Author: Feng Dai
Journal: Proc. Amer. Math. Soc. 134 (2006), 2931-2941
MSC (2000): Primary 41A15, 41A17; Secondary 41A05, 46E22
Published electronically: April 11, 2006
MathSciNet review: 2231617
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that second-order results can be attained by the generalized 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.

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  • [BD] G. Brown and F. Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Funct. Anal. 220 (2005), no. 2, 401-423. MR 2119285
  • [BDS] G. Brown, F. Dai and Y. Sh. Sun, Kolmogorov width of classes of smooth functions on the sphere $ \mathbb{S}\sp {d-1}$, J. Complexity 18 (2002), no. 4, 1001-1023. MR 1933699 (2003g:41039)
  • [LS] Q. T. Le Gia and I. H. Sloan, The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions, Constr. Approx. 17 (2001), no. 2, 249-265. MR 1814361 (2001m:41021)
  • [M1] H. N. Mhaskar, Polynomial operators and local smoothness classes on the unit interval, J. Approx. Theory 131 (2004), no. 2, 243-267.MR 2106540
  • [M2] H. N. Mhaskar, Weighted quadrature formulas and approximation by zonal function networks on the sphere, Preprint.
  • [MNW] H. N. Mhaskar and F.J. Narcowich and J.D. Ward, Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature, Math. Comp. 70 (2001), 1113-1130. MR 1710640 (2002a:41032)
  • [N] D. G. Northcott, Some inequalities between periodic functions and their derivatives, J. London Math. Soc. 14 (1939), 198-202.MR 0000417 (1:71c)
  • [Re1] M. Reimer, Hyperinterpolation on the sphere at the minimal projection order, J. Approx. Theory 104 (2000), no. 2, 272-286. MR 1761902 (2001c:41031)
  • [Re2] M. Reimer, Generalized hyperinterpolation on the sphere and the Newman-Shapiro operators, Constr. Approx. 18 (2002), no. 2, 183-203. MR 1890495 (2003a:41005)
  • [Sl] I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory 83 (1995), no. 2, 238-254. MR 1357589 (96h:41036)
  • [SW] I. H. Sloan and R. S. Womersley, Constructive polynomial approximation on the sphere, J. Approx. Theory 103 (2000), no. 1, 91-118. MR 1744380 (2000k:41009)
  • [St] S. B. Steckin, A generalization of some inequalities of S. N. Bernšte{\u{\i\/}}\kern.15emn (Russian), Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 1511-1514. MR 0024993 (9:579g)
  • [Sz] G. Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications 23, Fourth edition, American Mathematical Society, Providence, RI, 1975. MR 0372517 (51:8724)

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Additional Information

Feng Dai
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Keywords: Spherical polynomials, generalized hyperinterpolation, second-order moduli of smoothness, unit sphere
Received by editor(s): April 23, 2005
Published electronically: April 11, 2006
Additional Notes: The author was supported in part by the NSERC Canada under grant G121211001.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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