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

We study the order of best approximations Rₙ(p,α;f)=inf_gₖ,\,k≤ n||gₖ-f||p,α in the classical Bergman spaces Aᵖα(D) for the unit disk D by simple partial fractions gₖ (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<α<∞ , the bound Rₙ(p,α;f)=O(n-(α-p+1)/p) is established for any function f from the Hardy space Hᵖ(D) . The result is essentially sharp since the weak equivalence Rₙ(p,α;f) n-(α-p+1)/p is proved for any bounded analytic in D function f .