A multivariate Remez-type inequality with $\varphi$-concave weights

Michael I. Ganzburg ¹

¹ Department of Mathematics Hampton University Hampton, VA 23668, U.S.A.

Colloquium Mathematicum, Tome 147 (2017) no. 2, pp. 221-240

Résumé

Let $\varphi:[0,\infty)\to[0,\infty)$ be an increasing twice continuously differentiable function with a positive power index $\beta(\varphi):=\inf_{t \gt 0}(\varphi(t)/\varphi^\prime(t))^\prime$ and let $f:V\to[0,\infty)$ be concave on a convex body $V\subset\mathbb R^m$. In this paper we discuss the following Remez-type inequality for multivariate polynomials $P$ of degree $n$ on measurable sets $E\subseteq V$ equipped with a $\varphi$-concave measure $\mu(E):=\int_E\varphi(f(x))\,dx$: $$ \|P\|_{C(V)}\le T_n\biggl(\frac{2} {1-(1-\mu(E)/\mu(V))^{\beta(\varphi)/(1+m\beta(\varphi))}}-1\bigg)\|P\|_{C(E)}, $$ where $T_n$ is the Chebyshev polynomial of degree $n$. In addition, we describe the classes of all extremal measures $\mu$, bodies $V$, sets $E$, and polynomials $P$ for this inequality.

Détail

DOI : 10.4064/cm6884-7-2016

Keywords: varphi infty infty increasing twice continuously differentiable function positive power index beta varphi inf varphi varphi prime prime infty concave convex body subset mathbb paper discuss following remez type inequality multivariate polynomials degree measurable sets subseteq equipped varphi concave measure int varphi biggl frac beta varphi beta varphi bigg where chebyshev polynomial degree addition describe classes extremal measures bodies sets polynomials inequality