We study the geometric and approximation properties of Marcinkiewicz spaces and Stepanoff spaces, $1 \leq p < \infty$, as well as others where translations are not isometric but bounded (the bounded $p$-mean spaces) or even unbounded ($\Mean0$). We construct a function $f$ that belongs to these spaces and has the unusual and unexpected property that all approximate identities $\phi_\varepsilon*f$ converge to $f$ pointwise but they never converge in norm.
Andreano, F., Picardello, A.m. (2009). Approximate identities on some homogeneous Banach spaces. MONATSHEFTE FÜR MATHEMATIK, 158, 235-246 [10.1007/s00605-009-0106-2].
Approximate identities on some homogeneous Banach spaces
PICARDELLO, ANGELO MASSIMO
2009-01-01
Abstract
We study the geometric and approximation properties of Marcinkiewicz spaces and Stepanoff spaces, $1 \leq p < \infty$, as well as others where translations are not isometric but bounded (the bounded $p$-mean spaces) or even unbounded ($\Mean0$). We construct a function $f$ that belongs to these spaces and has the unusual and unexpected property that all approximate identities $\phi_\varepsilon*f$ converge to $f$ pointwise but they never converge in norm.File in questo prodotto:
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