What question did this study set out to answer?

This research aims to analyze the approximation of functions in Hardy spaces using logarithmic derivatives of polynomials.

June 18, 2026Open Access

Estimates for Approximation of Functions from Hardy Spaces H^p (D) by Logarithmic Derivatives of Polynomials, Whose Zeros Lie on the Unit Circle

Key Points

This research aims to analyze the approximation of functions in Hardy spaces using logarithmic derivatives of polynomials.
Studied best approximation order in Bergman spaces for functions from Hardy space.
Defined approximation using logarithmic derivatives of polynomials with poles on the unit circle.
Established bounds for the approximation order based on specific parameters (p and α).
Found that the best approximation rate is O(n^{-(α-p+1)/p}) for functions in Hardy spaces.
Proved that the result is sharp with the equivalence R_{n}(p,α;f) asymptotically equal to n^{-(α-p+1)/p}.
Validated findings for any bounded analytic function in the unit disk.

Abstract

We study the order of best approximations R₍ (p, ;f) =₆_₊, \, k n||g₊-f||₏, in the classical Bergman spaces A^p_ (D) for the unit disk D by simple partial fractions g₊ (i. e. , the logarithmic derivatives of polynomials of degree k n) with all poles on the unit circle T. For the case 0 p-1<<, the bound R₍ (p, ;f) =O (n^- (-p+1) /p) is established for any function f from the Hardy space H^p (D). The result is essentially sharp since the weak equivalence R₍ (p, ;f) n^- (-p+1) /p is proved for any bounded analytic in D function f.

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