Approximate identities on some homogeneous Banach spaces |
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Authors: | F Andreano M A Picardello |
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Institution: | 1.Department of Computer Sciences,University of Insubria,Varese,Italy;2.Department of Mathematics,University of Rome “Tor Vergata”,Rome,Italy |
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Abstract: | We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L p functions where translations are isometries, namely Marcinkiewicz spaces \({\mathcal{M}^{p}}\) and Stepanoff spaces \({\mathcal{S}^p}\), 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M p ) or even unbounded (\({M^{p}_{0}}\)). We construct a function f that belongs to these spaces and has the property that all approximate identities \({\phi_\varepsilon * f}\) converge to f pointwise but they never converge in norm. |
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