Weighted Approximation of Functions on the Unit Sphere
By Yuan Xu. 2003
81 citations recorded by the source corpus.
Abstract The direct and inverse theorems are established for the best approximation in the weighted Lp space on the unit sphere of Rd+1, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a K-functional defined using the power of the spherical h-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of Rd.
Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/4117424f8c093a0d5b71990b730d544637b21dd1
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In Applied Mathematics
- Pencho Petrushev
- Yuan Xu
- Feng Dai
- Francis J. Narcowich
- Joseph D. Ward
- Daryl Geller
- Azita Mayeli
- Gerard Kerkyacharian
In Numerical Analysis
In Statistics and Probability
Citations in the corpus
- Multivariate polynomial inequalities with respect to doubling weights and A∞ weights (2006, methodology): “We refer to [24,25] for Fourier analysis on B .”
- Decomposition of weighted Triebel–Lizorkin and Besov spaces on the ball (2007, unknown): “By the Jackson type estimate from [23], for any k ≥ 1, En(φ)2 ≤ ckn‖D μφ‖2 ≤ cn‖D μφ‖∞ ≤ cn ∑”
- Localized Polynomial Frames on the Ball (2006, background): “A result of the same nature holds true for more general weight functions of the form h(2)κ(x)(1 − |x|2)μ−1/2, where hκ(x) is some function invariant under a finite reflection group, see [20].”
- Localized Tight Frames on Spheres (2006, background): “Much work [11, 20, 22, 23, 27] has been done on estimating this error for various smoothness classes and spaces.”