On local RBF approximation (Q695650)

The paper deals with the problem of local approximation result for Radial Basis Function (RBF) approximation with the goal of effectively placing centers at varying resolutions. \textit{R. A. DeVore} and \textit{A. Ron} [Trans. Am. Math. Soc. 362, No. 12, 6205--6229 (2010; Zbl 1215.41004)], have established powerful local error estimates for kernel based approximation. They developed a satisfactory notion of local density the majorant-expressed as a function over a domain containing the centers. The authors here establish an alternative, equivalent characterization of the majorant of DeVore and Ron [loc. cit.], that involves a global compatibility condition and gives a self-contained development of their point wise error estimates that explicitly uses this compatibility condition. It is also shown that how the compatibility condition may be used to place centers non-uniformly. Lastly, a point wise comparison of results of DeVore and Ron (loc. cit.) to earlier results of \textit{Z.-M. Wu} and \textit{R. Schaback} [IMA J. Numer. Anal. 13, No. 1, 13--27 (1993; Zbl 0762.41006)] and of \textit{J. Duchon} [RAIRO, Anal. Numér. 12, 325-334 (1978; Zbl 0403.41003)] is given.