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Mathematics of Computation

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Modified Taylor reproducing formulas and h-p clouds

Author: Carlos Zuppa
Journal: Math. Comp. 77 (2008), 243-264
MSC (2000): Primary 41A17, 41A10; Secondary 65N15, 65N30
Published electronically: July 26, 2007
MathSciNet review: 2353952
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Abstract: We study two different approximations of a multivariate function $ f$ by operators of the form $ \sum_{i=1}^{N}\mathcal{T}_{r}[f,x_{i}](x)\,\mathcal{W} _{i}(x)$, where $ \{\mathcal{W}_{i}\}$ is an $ m$-reproducing partition of unity and $ \mathcal{T}_{r}[f,x_{i}](x)$ are modified Taylor polynomials of degree $ r$ expanded at $ x_{i}$. The first approximation was introduced by Xuli (2003) in the univariate case and generalized for convex domains by Guessab et al. (2005). The second one was introduced by Duarte (1995) and proved in the univariate case. In this paper, we first relax the Guessab's convexity assumption and we prove Duarte's reproduction formula in the multivariate case. Then, we introduce two related reproducing quasi-interpolation operators in Sobolev spaces. A weighted error estimate and Jackson's type inequalities for h-p cloud function spaces are obtained. Last, numerical examples are analyzed to show the approximative power of the method.

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Additional Information

Carlos Zuppa
Affiliation: Departamento de Matemáticas, Universidad Nacional de San Luis, San Luis 5700, Argentina

Keywords: Reproducing formulas, \textit{h-p} clouds method, error estimates
Received by editor(s): August 9, 2005
Received by editor(s) in revised form: October 24, 2006
Published electronically: July 26, 2007
Additional Notes: Partially supported by CyT-FCFMN-UNSL Grant 22/F330. The author wishes to thank the anonymous reviewers for their valuable suggestions.
Article copyright: © Copyright 2007 American Mathematical Society