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Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
The Uniform Norm of Hyperinterpolation on the Unit Sphere in an Arbitrary Number of Dimensions
, 2000
"... In this paper, we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S r1 # IR r . The hyperinterpolation approximation Ln f , where f # C(S r1 ), is derived from the exact L 2 orthogonal projection #n f onto the space P r n (S r1 ) ..."
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Cited by 5 (2 self)
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In this paper, we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S r1 # IR r . The hyperinterpolation approximation Ln f , where f # C(S r1 ), is derived from the exact L 2 orthogonal projection #n f onto the space P r n (S r1 ) of spherical polynomials of degree n or less, with the Fourier coe#cients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree # 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional "quadrature regularity" assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n r/21 ), which is the same as that of the orthogonal projection #n , and best possible among all linear projections onto P r n (S r1 ). Key words: hyperinterpolation, interpolation, reproducing kernel, unit sphere. AMS Subject Classifica...