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Sharp Constants of Approximation Theory. I. Multivariate Bernstein–Nikolskii Type Inequalities

Abstract

Given a centrally symmetric convex body \(V\subset {{\mathbb {R}}}^m\), let \({{\mathcal {T}}}_{aV}\) be the set of all trigonometric polynomials with the spectrum in \(aV,\,a>0\), and let \(B_V\) be the set of all entire functions of exponential type with the spectrum in V. We discuss limit relations between the sharp constants in the multivariate Bernstein–Nikolskii inequalities defined by

$$\begin{aligned}&P_{p,q,D_N,a,V}:=a^{-N-m/p+m/q}\sup _{T\in {{\mathcal {T}}}_{aV}\setminus \{0\}}\frac{\Vert D_N(T)\Vert _{L_q\left( Q_{\pi }\right) }}{\Vert T\Vert _{L_p\left( Q_{\pi }\right) }},\\&E_{p,q,D_N,V}:= \sup _{f\in (B_V\cap L_p({{\mathbb {R}}}^m))\setminus \{0\}}\frac{\Vert D_N(f)\Vert _{L_q({{\mathbb {R}}}^m)}}{\Vert f\Vert _{L_p({{\mathbb {R}}}^m)}}, \end{aligned}$$

where \(0<p\le q\le {\infty },\,Q_{\pi }=\{x\in {{\mathbb {R}}}^m: \vert x_j\vert \le \pi ,\,1\le j\le m\},\) and \( D_N=\sum _{\vert {\alpha }\vert =N}b_{\alpha }D^{\alpha }\) is a differential operator with constant coefficients. We prove that

$$\begin{aligned} E_{p,q,D_N,V}\le \liminf _{a\rightarrow {\infty }}P_{p,q,D_N,a,V},\qquad E_{p,{\infty },D_N,V}= \lim _{a\rightarrow {\infty }}P_{p,{\infty },D_N,a,V}. \end{aligned}$$

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Correspondence to Michael I. Ganzburg.

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Ganzburg, M.I. Sharp Constants of Approximation Theory. I. Multivariate Bernstein–Nikolskii Type Inequalities. J Fourier Anal Appl 26, 11 (2020) doi:10.1007/s00041-019-09720-x

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Keywords

  • Sharp constants
  • Multivariate Bernstein–Nikolskii inequality
  • Trigonometric polynomials
  • Entire functions of exponential type
  • Multivariate Levitan’s polynomials

Mathematics Subject Classification

  • Primary 41A17
  • 41A63
  • Secondary 26D05
  • 26D10