Comparison of the best approximation of holomorphic functions from Hardy space

Аbdullayev, F. G.; Savchuk, V. V.; Şimşek, Dağıstan

doi:10.22436/jnsa.012.07.01

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2022

2023

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“…The main reason why BAP operators are of special interest is that for a given f ∈ H q the convolution norm K n * f q , for a suitable K n , turns out to be a sharp lower bound for the best approximation E n (f ) q . For example, it was shown in [3] and [4] that the operator T n = K n * , where…”

Section: Introductionmentioning

confidence: 99%

Best Approximation-Preserving Operators over Hardy Space

Abdullayev¹,

Savchuk²,

Savchuk³

2022

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Let Tn be the linear Hadamard convolution operator acting over Hardy space H q , 1 ≤ q ≤ ∞. We call Tn a best approximation-preserving operator (BAP operator) if Tn(en) = en, where en(z) := z n , and if Tn(f ) q ≤ En(f )q for all f ∈ H q , where En(f )q is the best approximation by algebraic polynomials of degree a most n − 1 in H q 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 fn + c f N ≤ En(f )∞, where c > 0 and n < N , holds for every f ∈ H ∞ iff c ≤ 1 2 and N ≥ 2n + 1.

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Section: Introductionmentioning

confidence: 99%