Optimal convergence of minimum norm approximations inH _p

Summary

Let 1<p≦∞, and letH _p (U) denote the family of all functionsf that are analytic in the unit discU such that

$$\parallel f\parallel _p = \mathop {\lim }\limits_{r \to 1^ - } \left( {\frac{1}{{2\pi }}\int\limits_0^{2\pi } {|f(re^{i\theta } )|} ^p d\theta } \right)^{1/p}< \infty .$$

Set

$$\sigma _n = \mathop {\inf }\limits_{w_j ,x_j } \mathop {\sup }\limits_{f \in H^p (U),\parallel f\parallel p = 1} \left| {\int\limits_{ - 1}^1 {f(x)dx - \sum\limits_{j = 1}^n {w_j f(x_j )} } } \right|.$$

It is shown that given any ε>0, there exists an integern(ε)≧0, such that ifn>n(ε) andq=p/(p−1), then

$$\exp \left\{ { - \left( {5^{\tfrac{1}{2}} \pi + \varepsilon } \right)n^{\tfrac{1}{2}} } \right\} \leqq \sigma _n \leqq \exp \left\{ { - \left( {\frac{\pi }{{(2q)^{\tfrac{1}{2}} }} - \varepsilon } \right)n^{\tfrac{1}{2}} } \right\}.$$

LetH ^* _p (U) denote the family of all functionsf such thatg∈H _p (U), whereg(z)=f(z)/(1−z ²), and whereH ^* _p (U) is normed by ‖f‖ ^* _p =‖g‖ _p ‖,‖g‖ _p ‖ being defined as above. Let {T _n (f)} ^∞ _n=1 be an approximation scheme defined by

$$T_n (f)(z) = \sum\limits_{j = 1}^n {f(x_j )\phi _{n,j} (z),f \in H_p^* (U),} $$

where φ _n,j is analytic inU, and such that ‖T _n(f)‖ ^* _p ≦C‖f‖ ^* _p , whereC>0, but independent ofn. Then given any ε>0, there exists an integern(ε)≧0, such that whenevern>n(ε), then

$$\begin{gathered} \exp \left\{ { - (5^{\tfrac{1}{2}} \pi + \varepsilon )} \right.n^{\tfrac{1}{2}} \} \leqq \mathop {\inf }\limits_{Tn} \mathop {\sup }\limits_{f \in H_p^* (U),\parallel f\parallel _p^* = 1} \mathop {\sup }\limits_{ - 1< x< 1} |f(x) - T_n (f)(x)| \hfill \\ \leqq \exp \{ - \left. {\left( {\frac{\pi }{{2q^{\tfrac{1}{2}} }} - \varepsilon } \right)n^{\tfrac{1}{2}} } \right\}. \hfill \\ \end{gathered} $$