Optimal approximation order of piecewise constants on convex partitions

Abstract

We prove that the error of the best nonlinear L p -approximation by piecewise constants on convex partitions is O N − 2 d+1 , where N is the number of cells, for all functions in the Sobolev space W 2 q (Ω) on a cube Ω ⊂ R d , d 2, as soon asis achieved on a polyhedral partition obtained by anisotropic refinement of an adaptive dyadic partition. Further estimates of the approximation order from the above and below are given for various Sobolev and Sobolev-Slobodeckij spaces W r q (Ω) embedded in L p (Ω), some of which also improve the standard estimate O N − 1 d known to be optimal on isotropic partitions.

Spectral dimensions of Krein-Feller operators in higher dimensions

“…In following we use the terminology as in [DKS20]. Let Ξ 0 be a finite partition of Q of dyadic cubes from D. We say a partition Ξ of Q is an elementary extension of Ξ 0 , if it can be obtained from by uniformly splitting some of its cubes into 2 d equal sized disjoint cubes lying in D with half side length.…”