{
  "id": "2011.07619",
  "version": "v1",
  "published": "2020-11-15T20:01:26.000Z",
  "updated": "2020-11-15T20:01:26.000Z",
  "title": "Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions",
  "authors": [
    "Anatoliy Serdyuk",
    "Ulyana Hrabova"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We establish the exact-order estimates of uniform approximations by the Zygmund sums $Z^{s}_{n-1}$ of $2\\pi$-periodic continuous functions $f$ from the classes $C^{\\psi}_{\\beta,p}$.These classes are defined by the convolutions of functions from the unit ball in the space $L_{p}$, $1\\leq p<\\infty$, with generating fixed kernels $\\Psi_{\\beta}(t)=\\sum_{k=1}^{\\infty}\\psi(k)\\cos\\left(kt+\\frac{\\beta\\pi}{2}\\right)$, $\\Psi_{\\beta}\\in L_{p'}$, $\\beta\\in \\mathbb{R}$, $1/p+1/p'=1$. We additionally assume that the product $\\psi(k)k^{s+1/p}$ is generally monotonically increasing with the rate of some power function, and, besides, for $1< p<\\infty$ it holds that $\\sum_{k=n}^{\\infty}\\psi^{p'}(k)k^{p'-2}<\\infty$, and for $p=1$ the following condition is true $\\sum_{k=n}^{\\infty}\\psi(k)<\\infty$.It is shown that under these conditions Zygmund sums $Z^{s}_{n-1}$ and Fejer sums \\linebreak$\\sigma_{n-1}=Z^{1}_{n-1}$ realize the order of the best uniform approximations by trigonometric polynomials of these classes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-15T20:01:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J60",
      "F.2.2",
      "I.2.7"
    ],
    "keywords": [
      "periodic functions",
      "convolutions",
      "conditions zygmund sums",
      "best uniform approximations",
      "power function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}