The Nikolskii Constant in Arbitrary Dimension

Abstract

We study the radial extremal function $\phi (|\cdot |)$ arising in the problem of finding the sharp Nikolskii constant ${\mathcal{C}}_d$ in ${PW}_1^1({\mathbb{R}}^d)$ for arbitrary dimension $d\ge 1$. We prove a factorization $\phi ={\Phi }_1{\Phi }_2$, where ${\Phi }_1$ and ${\Phi }_2$ are entire functions of exponential type $1/2$ satisfying a functional equation and second-order differential equations with polynomial coefficients. As a result, the original extremal problem is reduced to a one-dimensional spectral problem depending on at most $d+1$ parameters. We also obtain a zeta interpretation of the coefficients of the polynomial appearing in the functional equation and a multiplicative equilibrium condition for the zeros of the extremal function. These results can be used to construct several algorithms for computing ${\mathcal{C}}_d$.

22 pages