The Nikolskii constant in odd dimensions

Abstract

Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $Φ$ of exponential type $1/2$ such that $\varphi_d(z)=Φ(z)Φ(-z)$, and $Φ$ satisfies a quadratic functional equation and a second-order linear differential equation with polynomial coefficients. Thus, the problem of finding the extremal function is reduced to a spectral problem with finitely many parameters. This result extends a recent one-dimensional result, but uses a different method. For example, in dimension $d=3$ it leads to a seven-diagonal spectral scheme that allows us to compute $\mathcal C_3$ to high accuracy. Even dimensions remain open within this approach.

Disclosure

“rities at ±1, in particular with factors (1−t2 )±1/2 . It remains to understand whether such a replacement leads to a finite-dimensional spectral problem analogous to the odd-dimensional case. Acknowledgments. The author is grateful to the AI model used in the preparation of this paper for its assistance. References [1] L. V. Ahlfors, Complex analysis, 3rd ed., McGraw–Hill, New York, 1979. [2] A. Bondarenko, J. Ortega-Cerdà, D. Radchenko”

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2026-08-23T05:17:51.028061Z

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