{
  "id": "1601.04483",
  "version": "v1",
  "published": "2016-01-18T11:55:35.000Z",
  "updated": "2016-01-18T11:55:35.000Z",
  "title": "Polynomial approximations to continuous functions and stochastic compositions",
  "authors": [
    "Takis Konstantopoulos",
    "Linglong Yuan",
    "Michael A. Zazanis"
  ],
  "comment": "21 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f \\in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k(n)$ to a polynomial $f$ when $k(n)/n$ tends to a constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-18T11:55:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "41A10",
      "41-01",
      "60H30"
    ],
    "keywords": [
      "stochastic compositions",
      "continuous function",
      "polynomial approximations",
      "stochastic process",
      "markov chain theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160104483K"
    }
  }
}