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Rademacher functions in Cesàro type spaces. (English) Zbl 1202.46031

Summary: The Rademacher sums are investigated in the Cesàro spaces \(\text{Ces}_p\) \((1\leq p\leq \infty)\) and in the weighted Korenblyum-Kreĭn-Levin spaces \(K_{p, w}\) on \([0, 1]\). They span \(l_2\) in \(\text{Ces}_p\) for any \(1\leq p<\infty\) and in \(K_{p, w}\) if and only if the weight \(w\) is larger than \(t \log_2^{p/2} ({2}/{t})\) on \((0, 1)\). Moreover, the span of the Rademachers is not complemented in \(\text{Ces}_p\) for any \(1\leq p<\infty\) or in \(K_{1, w}\) for any quasi-concave weight \(w\). In the case when \(p > 1\) and when \(w\) is such that the span of the Rademacher functions is isomorphic to \(l_2\), this span is a complemented subspace in \(K_{p,w}\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B42 Banach lattices
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