{
  "id": "math/0301380",
  "version": "v1",
  "published": "2003-01-31T20:27:39.000Z",
  "updated": "2003-01-31T20:27:39.000Z",
  "title": "An essay on some problems of approximation theory",
  "authors": [
    "A. G. Ramm"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Several questions of approximation theory are discussed: 1) can one approximate stably in $L^\\infty$ norm $f^\\prime$ given approximation $f_\\delta, \\parallel f_\\delta - f \\parallel_{L^\\infty} < \\delta$, of an unknown smooth function $f(x)$, such that $\\parallel f^\\prime (x) \\parallel_{L^\\infty} \\leq m_1$? 2) can one approximate an arbitrary $f \\in L^2(D), D \\subset \\R^n, n \\geq 3$, is a bounded domain, by linear combinations of the products $u_1 u_2$, where $u_m \\in N(L_m), m=1,2,$ $L_m$ is a formal linear partial differential operator and $N(L_m)$ is the null-space of $L_m$ in $D$, $3) can one approximate an arbitrary $L^2(D)$ function by an entire function of exponential type whose Fourier transform has support in an arbitrary small open set? Is there an analytic formula for such an approximation? N(L_m) := \\{w: L_m w=0 \\hbox{in\\} D\\}$?",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-01-31T20:27:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30D20",
      "35R25",
      "35J10",
      "35J05",
      "65M30"
    ],
    "keywords": [
      "approximation theory",
      "formal linear partial differential operator",
      "unknown smooth function",
      "approximate",
      "linear combinations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......1380R"
    }
  }
}