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Maryna S. Viazovska

  1. Bernstein type inequality in monotone rational approximation.

    Thu, 09/23/2010 - 15:02 — SomNambul
    Authors: Andriy V. Bondarenko, Maryna S. Viazovska
    Subjects: Numerical Analysis
    Abstract

    The following analog of Bernstein inequality for monotone rational functions
    is established: if $R$ is an increasing on $[-1,1]$ rational function of degree
    $n$, then $$ R'(x)<\frac{9^n}{1-x^2}\|R\|,\quad x\in (-1,1). $$ The exponential
    dependence of constant factor on $n$ is shown, with sharp estimates for odd
    rational functions.

  2. On optimal asymptotic bounds for spherical designs.

    Thu, 09/23/2010 - 15:02 — SomNambul
    Authors: Andriy V. Bondarenko, Danylo V. Radchenko, Maryna S. Viazovska
    Subjects: Metric Geometry
    Abstract

    For each $N\ge c_dt^d$ we prove the existence of a spherical $t$-design on
    the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending
    only on $d$. This result proves the well-known conjecture of Korevaar and
    Meyers concerning an optimal order of minimal number of points in a spherical
    $t$-design on $S^d$ for a fixed $d$.

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