Best approximation-preserving operators over Hardy space

ABDULLAYEV F., Savchuk V., Savchuk M.

Analysis and Mathematical Physics, vol.13, no.4, 2023 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 13 Issue: 4
  • Publication Date: 2023
  • Doi Number: 10.1007/s13324-023-00825-7
  • Journal Name: Analysis and Mathematical Physics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Keywords: Best approximation, Cauchy inequality, Hadamard product, Hardy space, Landau inequality
  • Uşak University Affiliated: No

Abstract

Let Tn be the linear Hadamard convolution operator acting over Hardy space Hq , 1 ≤ q≤ ∞ . We call Tn a best approximation-preserving operator (BAP operator) if Tn(en) = en , where en(z) : = zn, and if ‖ Tn(f) ‖ q≤ En(f) q for all f∈ Hq , where En(f) q is the best approximation by algebraic polynomials of degree a most n- 1 in Hq space. We give necessary and sufficient conditions for Tn to be a BAP operator over H∞ . We apply this result to establish an exact lower bound for the best approximation of bounded holomorphic functions. In particular, we show that the Landau-type inequality | f^ n| + c| f^ N| ≤ En(f) ∞ , where c> 0 and n< N , holds for every f∈ H∞ iff c≤12 and N≥ 2 n+ 1 .