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Exact Asymptotics of the Multidimensional Nikolskii Constant

  • D. V. Gorbachev
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Abstract

We prove the exact asymptotics of the multidimensional normalized L1 Nikolskii constant ℒ*(d)=(π/2+o(1))2−d, d→∞. In addition, for each fixed dimension we obtain asymptotics for the positive zeros of the extremal function φd, and determine their limiting distribution as d→∞. The lower bound follows from the construction of an admissible function and its asymptotic analysis. For the upper bound, we represent xd+1φd(x) as the product of two solutions of the equation u''+Vd(x)u=0, compare this equation with a Bessel model, and analyze relative canonical products.

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Cite

BibTeXRIS
BibTeX
@misc{gorbachev2026exact,
  title = {{Exact Asymptotics of the Multidimensional Nikolskii Constant}},
  author = {D. V. Gorbachev},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06240},
  eprint = {2609.06240},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06240},
}
RIS
TY  - UNPB
TI  - Exact Asymptotics of the Multidimensional Nikolskii Constant
AU  - D. V. Gorbachev
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06240
UR  - https://arxiv.org/abs/2609.06240
AB  - We prove the exact asymptotics of the multidimensional normalized L^1 Nikolskii constant ℒ^*(d)=(π/2+o(1))2^−d, d→∞. In addition, for each fixed dimension we obtain asymptotics for the positive zeros of the extremal function φ_d, and determine their limiting distribution as d→∞. The lower bound follows from the construction of an admissible function and its asymptotic analysis. For the upper bound, we represent x^d+1φ_d(x) as the product of two solutions of the equation u''+V_d(x)u=0, compare this equation with a Bessel model, and analyze relative canonical products.
ER  -