Exact Asymptotics of the Multidimensional Nikolskii Constant

Abstract

We prove the exact asymptotics of the multidimensional normalized $L^1$ Nikolskii constant \[ \mathcal L^*(d)=\Bigl(\fracπ{2}+o(1)\Bigr)2^{-d}, \quad d\to\infty. \] In addition, for each fixed dimension we obtain asymptotics for the positive zeros of the extremal function ${\phi }_d$, and determine their limiting distribution as $d\to \infty$. The lower bound follows from the construction of an admissible function and its asymptotic analysis. For the upper bound, we represent $x^{d+1}{\phi }_d(x)$ as the product of two solutions of the equation $u^{\prime \prime }+V_d(x)u=0$, compare this equation with a Bessel model, and analyze relative canonical products.

27 pages