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Best Polynomial Approximation on the Unit Sphere and the Unit Ball

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Approximation Theory XIV: San Antonio 2013

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 83))

Abstract

This is a survey on best polynomial approximation on the unit sphere and the unit ball. The central problem is to describe the approximation behavior of a function by polynomials via smoothness of the function. A major effort is to identify a correct gadget that characterizes smoothness of functions, either a modulus of smoothness or a \(K\)-functional, both of which are often equivalent. We concentrate on characterization of best approximations, given in terms of direct and converse theorems, and report several moduli of smoothness and \(K\)-functionals, including recent results that give a fairly satisfactory characterization of best approximation by polynomials for functions in \(L^p\) spaces, the space of continuous functions, and Sobolev spaces.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Yuan Xu

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  1. Yuan Xu
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Correspondence to Yuan Xu .

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Editors and Affiliations

  1. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois, USA

    Gregory E. Fasshauer

  2. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Xu, Y. (2014). Best Polynomial Approximation on the Unit Sphere and the Unit Ball. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_22

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