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On an extremal scale of approximation spaces. (English) Zbl 1033.46021
Let \((X, \| \cdot \| )\) be a quasinormed Abelian group and \(\{G_ n : n\in \mathbb N_ 0\}\) a sequence of subsets of \(X\) satisfying the conditions: (1) \(G_ 0 = \{0\}\); (2) \(G_ n \subset G_{n+1}\), \(n\in \mathbb N_ 0\); (3) \(G_ n\pm G_ m \subset G_{n+m}.\) The approximation error with respect to \(G_{n-1}\) of an element \(f\in X\) is defined by \(E_ n(f) = \inf\{\| f-g\| : g\in G_{n-1}\}.\) For \(\alpha > 0\) and \(0< q \leq \infty\), define the approximation space \(X^ \alpha_ q\) as the set of all \(f\in X\) for which the quasinorm \(\| f\| ^ q_{\alpha ,q} = \sum_{n=1}^ \infty((n^\alpha E_ n(f))^ qn^{-1}\) (with the usual convention for \(q=\infty\)) is finite. These spaces, introduced and studied by P. L. Butzer and K. Scherer [Approximationsprozesse und Interpolationsmethoden, Hochschulskripten, Mannheim-Zürich: Bibliographisches Institut (1968; Zbl 0177.08501)], include, for appropriate choices of the sets \(G_ n\), important classes of spaces, as, for instance, the Besov spaces and the approximation operator ideals.
The authors of this paper consider the spaces \(X^{(0,\gamma )}_ q\) of all \(f\in X\) for which the logarithmic quasinorm \(\| f\| ^ q_{(0,\gamma),q} = \sum_{n=1}^ \infty((1+\log n)^ \gamma E_ n(f))^ qn^{-1}\) is finite. For \(\gamma = 0\) and \(0< q < \infty\), one obtains the spaces \(X_ q:= X^{(0,0)}_ q,\) studied by F. Cobos and M. Milman [Numer. Funct. Anal. Optim. 11, 11-31 (1990; Zbl 0729.41033)], as a limit case (for \(\alpha = 0\)) of the spaces \(X^ \alpha_ q.\) They study the behavior of these spaces with respect to reiteration and interpolation. For instance, \((X_ q^{(0,\gamma)})_ r^{(0,\delta)} = X_ r^{(0,\gamma + \delta +1/q)}\) (Theorem 2), and \((X,X^{(0,\gamma )})_{\theta ,q} = X_ q^{(0,\delta)},\) where \( \delta = \theta (\gamma +1/p)-1/q\) (Theorem 4).
Applications are given to Lorentz-Zygmund operator ideals and to Besov spaces with logarithmic weights.

MSC:
46B70 Interpolation between normed linear spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46B10 Duality and reflexivity in normed linear and Banach spaces
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