zbMATH — the first resource for mathematics

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.

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
Full Text: DOI