The Nikolskii constant in odd dimensions

Abstract

Let ${\phi }_d(|\cdot |)$ be the radial extremal function in the problem for the sharp Nikolskii constant ${\mathcal{C}}_d^{-1}=\inf \Vert f{\Vert }_1$ over functions $f\in PW{}_1^1({\mathbb{R}}^d)$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that ${\phi }_d(z)=\Phi (z)\Phi (-z)$, and $\Phi$ satisfies a quadratic functional equation and a second-order linear differential equation with polynomial coefficients. Thus, the problem of finding the extremal function is reduced to a spectral problem with finitely many parameters. This result extends a recent one-dimensional result, but uses a different method. For example, in dimension $d=3$ it leads to a seven-diagonal spectral scheme that allows us to compute ${\mathcal{C}}_3$ to high accuracy. Even dimensions remain open within this approach.

31 pages