Width of the Hardy classes \(H^ p\) in weighted \(L_ q\) spaces
type: Item
instance of: scholarly article
title: Width of the Hardy classes \(H^ p\) in weighted \(L_ q\) spaces
zbMATH Open document ID: 0792.30025
DOI: 10.1007/BF01250559
author: O. G. Parfenov
published in: Mathematical Notes
publication date: 1994-02-08T00:00:00Z
review text: Let \(X,Y\) be normed linear spaces, \(x \in X \Rightarrow x \in Y(\| x \|_ X \neq \| x \|_ Y\), generally), \(B\) be the unit ball in \(X\), \(A \in L(X \to Y)\), \[ \begin{aligned} \lambda_ n(X,Y) & =\inf_ A \sup_{z \in B} \| z-Az \|_ Y,\dim AX \leq n,\\ d^ n(X,Y) & = \inf_{Y_ n} \sup_{z \in B} \inf_{W \in Y_ n} \| z-W \|_ Y,\;Y_ n \subset Y,\dim Y_ n \leq n,\\ d_ n(X,Y) & =\inf_{Y_{- n}} \sup_{z \in B \cap Y_{-n}} \| W \|_ Y,Y_{-n} \subset Y,\text{ codim } Y_{-n} \leq n.\end{aligned} \] The values \(\lambda_ n(X,Y)\), \(d^ n(X,Y)\), \(d_ n(X,Y)\) are called linear, Kolmogorov, Gel'fand widths, respectively, of the ball \(B\) in the metric of the space \(Y\). The widths have applications in the theory of approximation, theory of interpolation of linear operators, another questions. The width was calculated for some pairs of spaces. Let \(H^ p\) be the Hardy space in the unit circle, \(T_ r=\{z:| z |=r\}\), \(0<r<1\), \(L_ p (T_ r, \mu)\) be the Lebesgue space respecting finite Borel measure \(\mu\). The author proves that \[ \begin{multlined} \lambda_ n (H^ p,L_ q \bigl( T_ r,\mu) \bigr)=d^ n \bigl( H^ p,L_ q (T_ r,\mu) \bigr)=d_ n \bigl( H^ p,L_ q (T_ r,\mu) \bigr)=\\ G^{1/q} (\mu)r^ n,G (\mu)=\exp S \left( \ln {d \mu \over dm} \right) dm,\end{multlined} \] where \(m\) is the normalized Lebesgue measure on \(T_ r\), \(1 \leq q \leq p \leq \infty\), \[ \begin{multlined} \varlimsup_{n \to \infty} \lambda_ k \bigl( H^ p,L_ q (T_ r,\mu) \bigr) r^{-n} \leq G^{1/q} (\mu) (1-r^ s)^{- {1 \over s}},\;1 \leq p \leq 2 \leq q \leq \infty, \\ {1 \over p}={1 \over q} +{1 \over s},\;\mu=\varphi dm,\;\varphi \in C(T_ r).\end{multlined} \] {}.
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 484294
zbMATH Keywords: width of unit ball
zbMATH Keywords: Lebesgue space
MaRDI profile type: Publication
cites work: Q3922907
cites work: Q4115573
cites work: The n-width of sets of analytic functions
cites work: Optimal Sampling of Holomorphic Functions
description: scientific article; zbMATH DE number 484294