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Search: All articles in the CMB digital archive with keyword Sobolev space

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1. CMB 2011 (vol 54 pp. 566)

Zhou, Xiang-Jun; Shi, Lei; Zhou, Ding-Xuan
 Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter. Keywords:multivariate approximation, Sobolev spaces, non-uniform randomized sampling, high order Parzen windows, convergence ratesCategories:68T05, 62J02

2. CMB 1999 (vol 42 pp. 463)

Hofmann, Steve; Li, Xinwei; Yang, Dachun
 A Generalized Characterization of Commutators of Parabolic Singular Integrals Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1, \dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots \le\az_n$. Denote $|\az|=\az_1+\cdots+\az_n$. We characterize those functions $A(x)$ for which the parabolic Calder\'on commutator $$T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n} K(x-y)[A(x)-A(y)]f(y)\,dy$$ is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{-|\az|-1}K(x)$, $K$ is smooth away from the origin and satisfies a certain cancellation property. Keywords:parabolic singular integral, commutator, parabolic $\BMO$ sobolev space, homogeneous space, T1-theorem, symbolCategory:42B20