Best approximation-preserving operators over Hardy space

---

ABDULLAYEV F., Savchuk V., Savchuk M.

Analysis and Mathematical Physics, cilt.13, sa.4, 2023 (SCI-Expanded, Scopus)

• Yayın Türü: Makale / Tam Makale
• Cilt numarası: 13 Sayı: 4
• Basım Tarihi: 2023
• Doi Numarası: 10.1007/s13324-023-00825-7
• Dergi Adı: Analysis and Mathematical Physics
• Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
• Anahtar Kelimeler: Best approximation, Cauchy inequality, Hadamard product, Hardy space, Landau inequality
• Uşak Üniversitesi Adresli: Hayır

Özet

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 .