Best approximation of holomorphic functions from hardy space in terms of Taylor coefficients

Аbdullayev, F. G.; Abdullayev, G. A.; Savchuk, V. V.

doi:10.2298/fil1905417a

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Cited by 3 publications

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“…The inequality ( 14) is sharp on whole space H q in the following sense: equality in ( 14) for given n, as was shown in [9], is attained if and only if…”

Section: Applicationmentioning

confidence: 74%

Best Approximation-Preserving Operators over Hardy Space

Abdullayev¹,

Savchuk²,

Savchuk³

2022

Preprint

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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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“…The inequality ( 14) is sharp on whole space H q in the following sense: equality in ( 14) for given n, as was shown in [9], is attained if and only if…”

Section: Applicationmentioning

confidence: 74%