The Nikolskii Constant in Arbitrary Dimension

Abstract

We study the radial extremal function $\varphi(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $\varphi=\Phi_1\Phi_2$, 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 $\mathcal C_d$.