{
  "id": "1901.03907",
  "version": "v1",
  "published": "2019-01-12T22:04:24.000Z",
  "updated": "2019-01-12T22:04:24.000Z",
  "title": "On some properties of moduli of smoothness with Jacobi weights",
  "authors": [
    "K. A. Kopotun",
    "D. Leviatan",
    "I. A. Shevchuk"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as \\[ \\omega_{k,r}^\\varphi(f^{(r)},t)_{\\alpha,\\beta,p} :=\\sup_{0\\leq h\\leq t} \\left\\| {\\mathcal{W}}_{kh}^{r/2+\\alpha,r/2+\\beta}(\\cdot) \\Delta_{h\\varphi(\\cdot)}^k (f^{(r)},\\cdot)\\right\\|_p \\] where $\\varphi(x) = \\sqrt{1-x^2}$, $\\Delta_h^k(f,x)$ is the $k$th symmetric difference of $f$ on $[-1,1]$, \\[ {\\mathcal{W}}_\\delta^{\\xi,\\zeta} (x):= (1-x-\\delta\\varphi(x)/2)^\\xi (1+x-\\delta\\varphi(x)/2)^\\zeta , \\] and $\\alpha,\\beta > -1/p$ if $0<p<\\infty$, and $\\alpha,\\beta \\geq 0$ if $p=\\infty$. We show, among other things, that for all $m, n\\in N$, $0<p\\le \\infty$, polynomials $P_n$ of degree $<n$ and sufficiently small $t$, \\begin{align*} \\omega_{m,0}^\\varphi(P_n, t)_{\\alpha,\\beta,p} & \\sim t \\omega_{m-1,1}^\\varphi(P_n', t)_{\\alpha,\\beta,p} \\sim \\dots \\sim t^{m-1}\\omega_{1,m-1}^\\varphi(P_n^{(m-1)}, t)_{\\alpha,\\beta,p} & \\sim t^m \\left\\| w_{\\alpha,\\beta} \\varphi^{m} P_n^{(m)}\\right\\|_{p} , \\end{align*} where $w_{\\alpha,\\beta}(x) = (1-x)^\\alpha (1+x)^\\beta$ is the usual Jacobi weight. In the spirit of Yingkang Hu's work, we apply this to characterize the behavior of the polynomials of best approximation of a function in a Jacobi weighted $L_p$ space, $0<p\\le\\infty$. Finally we discuss sharp Marchaud and Jackson type inequalities in the case $1<p<\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-12T22:04:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smoothness",
      "properties",
      "yingkang hus work",
      "usual jacobi weight",
      "th symmetric difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}