{
  "id": "1608.04114",
  "version": "v1",
  "published": "2016-08-14T17:06:25.000Z",
  "updated": "2016-08-14T17:06:25.000Z",
  "title": "Approximation by polynomials in Sobolev spaces with Jacobi weight",
  "authors": [
    "Yuan Xu"
  ],
  "categories": [
    "math.CA",
    "math.NA"
  ],
  "abstract": "Polynomial approximation is studied in the Sobolev space $W_p^r(w_{\\alpha,\\beta})$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{\\alpha,\\beta}$. This requires simultaneous approximation of a function and its consecutive derivatives up to $s$-th order with $s \\le r$. We provide sharp error estimates given in terms of $E_n(f^{(r)})_{L^p(w_{\\alpha,\\beta})}$, the error of best approximation to $f^{(r)}$ by polynomials in $L^p(w_{\\alpha,\\beta})$, and an explicit construction of the polynomials that approximate simultaneously with the sharp error estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-14T17:06:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A10",
      "41A25",
      "42C05",
      "42C10",
      "33C45"
    ],
    "keywords": [
      "sobolev space",
      "sharp error estimates",
      "jacobi weight function",
      "th derivatives",
      "th order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}