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M.J.D. Powell's work in univariate and multivariate approximation theory and his contribution to optimization

by M. D. Buhmann and R. Fletcher

(Report number 1996-16)

Abstract
Since 1966, exactly 30 years ago, Mike Powell has published more than 40 papers in approximation theory, initially mostly on univariate approximations and then, focussing especially on radial basis functions, also on multivariate methods. A highlight of his work is certainly his book {\it Approximation theory and methods}, published by CUP in 1981, that summarizes and extends much of his work on $\ell_1$, $\ell_2$, $\ell_\infty$ theory and methods, splines, polynomial and rational approximation etc. It is still one of the best available texts on univariate approximation theory. In this short article we attempt to introduce part of Mike's work, with special emphasis on splines in one dimension on the one hand and radial basis functions on the other hand. Only a selection of his papers can be considered, and we are compelled to leave out all of his many software contributions, which for Mike are an integral part of his research work, be it for the purpose of establishing new or better methods for approximation or for making them more accessible to the general public through library systems. We subdivide this chapter into three parts ($\ell_1 / \ell_\infty$-approximation, rational approximation; splines; multivariate (radial basis function) approximation) although this is in variance with the spirit of many of Mike's articles which often establish beautiful links between different themes (e.g. optimization and $\ell_1$-approximation). As will be seen, many of the papers contain optimal results in the sense that constants in error estimates are best (or the best ones known), have also often surprising novelty and always clearly defined goals. One further important contribution that we cannot describe here is Mike's guidance for the seven dissertations in approximation theory that were written under his supervision. In a second chapter, Mike's contributions to optimization are reviewed with a special emphasis on the historical development of the subject and the impact of Mike's work on it.

Keywords:

@Techreport{BF96_199,
  author = {M. D. Buhmann and R. Fletcher},
  title = {M.J.D. Powell's work in univariate and multivariate approximation theory and his contribution to optimization},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1996-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1996/1996-16.pdf },
  year = {1996}
}

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