{
  "id": "2302.05308",
  "version": "v1",
  "published": "2023-02-10T15:10:42.000Z",
  "updated": "2023-02-10T15:10:42.000Z",
  "title": "Lower bounds on the measure of the support of positive and negative parts of trigonometric polynomials",
  "authors": [
    "Abdulamin Ismailov"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For a finite set of natural numbers $D$ consider a complex polynomial of the form $f(z) = \\sum_{d \\in D} c_d z^d$. Let $\\rho_+(f)$ and $\\rho_-(f)$ be the fractions of the unit circle that $f$ sends to the right($\\operatorname{Re} f(z) > 0$) and left($\\operatorname{Re} f(z) < 0$) half-planes, respectively. Note that $\\operatorname{Re} f(z)$ is a real trigonometric polynomial, whose allowed set of frequencies is $D$. Turns out that $\\min(\\rho_+(f), \\rho_-(f))$ is always bounded below by a numerical characteristic $\\alpha(D)$ of our set $D$ that arises from a seemingly unrelated combinatorial problem. Furthermore, this result could be generalized to power series, functions of several variables and multivalued algebraic functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-10T15:10:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower bounds",
      "negative parts",
      "real trigonometric polynomial",
      "natural numbers",
      "power series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}