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.