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<p>In this paper, we study the form of the constant <i>C</i> in the Bernstein–Nikolskii inequalities <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10157_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="301" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Vert f^{(s)}\Vert _q \lesssim C(s, p, q)\left\| f\right\| _p,\,0&lt;p&lt;q \leqslant \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <mi>q</mi> </msub> <mo>≲</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mfenced close="∥" open="∥"> <mi>f</mi> </mfenced> <mi>p</mi> </msub> <mo>,</mo> <mspace width="0.166667em" /> <mn>0</mn> <mo>&lt;</mo> <mi>p</mi> <mo>&lt;</mo> <mi>q</mi> <mo>⩽</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, for trigonometric polynomials and entire functions of exponential type. We obtain the optimal behavior of the constant with respect to the smoothness parameter <i>s</i>.</p>

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Bernstein–Nikolskii Inequalities: Optimality with Respect to the Smoothness Parameter

  • 期刊论文

  • 发表日期：

    2025年5月13日

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  • Michael I. Ganzburg,
  • Miquel Saucedo,
  • Sergey Tikhonov

摘要

In this paper, we study the form of the constant C in the Bernstein–Nikolskii inequalities \(\Vert f^{(s)}\Vert _q \lesssim C(s, p, q)\left\| f\right\| _p,\,0<p<q \leqslant \infty \) ‖ f ( s ) ‖ q ≲ C ( s , p , q ) f p , 0 < p < q ⩽ ∞ , for trigonometric polynomials and entire functions of exponential type. We obtain the optimal behavior of the constant with respect to the smoothness parameter s.