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arXiv · 2608.22578

The Nikolskii Constant in Arbitrary Dimension

D. V. Gorbachev
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Abstract

We study the radial extremal function $φ(|{\,\cdot\,}|)$ 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 $φ=Φ_1Φ_2$, where $Φ_1$ and $Φ_2$ 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 $\mathcal C_d$.

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D. V. Gorbachev. 2026-08-23. The Nikolskii Constant in Arbitrary Dimension. https://arxiv.org/abs/2608.22578

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