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On a limit class of approximation spaces. (English) Zbl 0729.41033
Let X be a quasi-normed abelian group and let \(G_ 0\subset G_ 1...G_ n\) be a sequence of subsets of X such that \(G_ n\pm G_ m\subset G_{n+m}\). For any \(q\in (0,\infty)\), the authors study the space \[ X_ q=\{f\in X:\;\| f\|_{X_ q}=[\sum^{\infty}_{n- 1}E_ n(f)^ qn^{-1}]^{1/q}<\infty \}, \] where \(E_ n(f)=\inf \{\| f-g\|:\) \(g\in G_{n-1}\}\). These spaces are limiting cases of the approximation spaces \(X^{\alpha}_ q\) studied by J. Peetre [Notas Mat. 39, 1-86 (1968; Zbl 0162.445)], A. Pietsch [J. Approximation Theory 32, 115-134 (1981; Zbl 0489.47008)], and others. The authors show that the family \(\{X_ q\}_{q>0}\) is closed under interpolation, and they characterize the spaces \(X_ q\) as extrapolation spaces between X and the space \[ X^ 1_ 1=\{f\in X:\;\| f\|_{X^ 1_ 1}=\sum^{\infty}_{n-1}E_ n(f)<\infty \}. \] After giving several sufficient conditions for boundedness of an operator J: \(X\times Y\to Z\) between approximation spaces, the authors give several applications to the operator ideals \({\mathcal L}_{\infty,q}\).
Reviewer: R.M.Aron (Kent)

MSC:
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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