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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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