1. CMB Online first
|On the Regularity of the Multisublinear Maximal Functions|
This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.
Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuity
2. CMB 2011 (vol 54 pp. 566)
|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 rates