Exact Asymptotics of the Multidimensional Nikolskii Constant

We prove the exact asymptotics of the multidimensional normalized $L^1$ Nikolskii constant $${\mathcal{L}}^{\ast }(d)=(\frac{\pi }{2}+o(1))2^{-d},d\to \mathrm{\infty }\mathrm{.}$$ 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 $d\to \mathrm{\infty }$. 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 $u^{\mathrm{\prime }\mathrm{\prime }}+V_d(x)u=0$, compare this equation with a Bessel model, and analyze relative canonical products.