Whitney Estimates for Convex Domains with Applications to Multivariate Piecewise Polynomial Approximation
| 详细信息 | 推荐本文 | |
- 作者：S. Dekel and D. Leviatan
- 关键词：Piecewise polynomialapproximation ; Nonlinear approximation ; Whitney estimates ; Johnstheorem
- 刊名：Foundations of Computational Mathematics
- 出版时间：November 2004
- 出版年：2004
- 期刊代码：54_1615-3375
- 类别：cp
- 卷：4
- 期：4
- 页码：345-368
- 数据来源：sp
摘要
We prove the following Whitney estimate. Given 0 < p \le \infty,r \in N, and d \ge 1, there exists a constant C(d,r,p),depending only on the three parameters, such that for everybounded convex domain \subset Rd, and each functionf \in Lp(),Er-1(f,)p \le C(d,r,p)r(f, diam())p,where Er-1(f,)p is the degree ofapproximation by polynomials of total degree, r – 1, andr(f,·)p is the modulus of smoothness of order r.Estimates like this can be found in the literature but withconstants that depend in an essential way on the geometry of thedomain, in particular, the domain is assumed to be a Lipschitzdomain and the above constant C depends on the minimalhead-angle of the cones associated with the boundary.The estimates we obtain allow us to extend to the multivariatecase, the results on bivariate Skinny B-spaces of Karaivanov andPetrushev on characterizing nonlinear approximation from nestedtriangulations. In a sense, our results were anticipated byKaraivanov and Petrushev.