ISSN:
0945-3245
Keywords:
AMS(MOS): 65D15
;
CR: 5.13
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 $$\begin{gathered} \exp \left\{ { - (5^{\tfrac{1}{2}} \pi + \varepsilon )} \right.n^{\tfrac{1}{2}} \} \leqq \mathop {\inf }\limits_{Tn} \mathop {\sup }\limits_{f \in H_p^* (U),\parallel f\parallel _p^* = 1} \mathop {\sup }\limits_{ - 1〈 x〈 1} |f(x) - T_n (f)(x)| \hfill \\ \leqq \exp \{ - \left. {\left( {\frac{\pi }{{2q^{\tfrac{1}{2}} }} - \varepsilon } \right)n^{\tfrac{1}{2}} } \right\}. \hfill \\ \end{gathered} $$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01432874
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