多项式和指数型整函数的Sharp Bernstein--Nikolskii不等式
Sharp Bernstein--Nikolskii inequalities for polynomials and entire functions of exponential type
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中文总结 AI 辅助
本文综述多项式和指数型整函数的Bernstein--Nikolskii不等式，聚焦尖锐常数与极值函数，涵盖经典与近期结果、加权多维版本及开放问题。
中文摘要 AI 辅助
我们综述了多项式和指数型整函数的Bernstein--Nikolskii不等式，特别关注尖锐常数和极值函数。我们讨论了经典结果和近期发展，包括多项式与指数型整函数的Nikolskii常数之间的极限关系、加权和多维版本，以及受限函数类上的不等式。特别关注困难情形$p=1$以及基于再生核、对偶性和极值逼近的方法。我们还描述了尖锐Bernstein--Nikolskii不等式与逼近论、Riemann zeta函数、球面设计、Turán--Delsarte型问题以及球堆积的联系。文中提出了若干关于尖锐常数、极值函数和多维推广的开放问题。
英文摘要
We survey Bernstein--Nikolskii inequalities for polynomials and entire functions of exponential type, with particular emphasis on sharp constants and extremal functions. We discuss both classical results and recent developments, including limit relations between Nikolskii constants for polynomials and entire functions of exponential type, weighted and multidimensional versions, and inequalities on constrained classes of functions. Special attention is given to the difficult case $p=1$ and to methods based on reproducing kernels, duality, and extremal approximation. We also describe connections of sharp Bernstein--Nikolskii inequalities with approximation theory, the Riemann zeta function, spherical designs, Turán--Delsarte type problems, and sphere packing. Several open problems concerning sharp constants, extremal functions, and multidimensional extensions are formulated.
