{
  "id": "2411.10135",
  "version": "v1",
  "published": "2024-11-15T12:14:32.000Z",
  "updated": "2024-11-15T12:14:32.000Z",
  "title": "On the rates of pointwise convergence for Bernstein polynomials",
  "authors": [
    "José A. Adell",
    "Daniel Cárdenas-Morales",
    "Antonio J. López-Moreno"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $f$ be a real function defined on the interval $[0,1]$ which is constant on $(a,b)\\subset [0,1]$, and let $B_nf$ be its associated $n$th Bernstein polynomial. We prove that, for any $x\\in (a,b)$, $|B_nf(x)-f(x)|$ converges to $0$ as $n\\rightarrow \\infty $ at an exponential rate of decay. Moreover, we show that this property is no longer true at the boundary of $(a,b)$. Finally, an extension to Bernstein-Kantorovich type operators is also provided",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-15T12:14:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A10",
      "41A25",
      "41A36",
      "60E05"
    ],
    "keywords": [
      "pointwise convergence",
      "th bernstein polynomial",
      "bernstein-kantorovich type operators",
      "exponential rate",
      "real function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}