{
  "id": "1402.5671",
  "version": "v1",
  "published": "2014-02-23T20:47:45.000Z",
  "updated": "2014-02-23T20:47:45.000Z",
  "title": "Best Polynomial Approximation on the Unit Sphere and the Unit Ball",
  "authors": [
    "Yuan Xu"
  ],
  "comment": "To appear in Approximation Theorey XIV: San Antonio 2013, G. Fasshauer and L. Schumaker eds., Springer",
  "categories": [
    "math.CA"
  ],
  "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, the two of which are often equivalent. We will 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-23T20:47:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "best polynomial approximation",
      "unit sphere",
      "unit ball",
      "best approximation",
      "smoothness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.5671X"
    }
  }
}