Abstract
The Rademacher series in rearrangement invariant function spaces “close” to the space L∞ are considered. In terms of interpolation theory of operators, a correspondence between such spaces and spaces of coefficients generated by them is stated. It is proved that this correspondence is one‐to‐one. Some examples and applications are presented.
Highlights
Let rk(t) = sign sin 2k−1π t (k = 1, 2, . . .) (1.1)be the Rademacher functions on the segment [0, 1]
For a Banach couple (X0, X1), x ∈ X0 + X1 and t > 0, we introduce the Peetre functional
Theorem 1.2 allows to find an r.i.s. that contains Rademacher series with coefficients belonging to an arbitrary interpolation space between l1 and l2
Summary
If (X0, X1) is a Banach couple, the space of the real -method of interpolation (X0, X1)E consists of all x ∈ X0 + X1 such that x = 2k, x; X0, X1 k E < ∞. t, y; X0, X1 ≤ t, x; X0, X1 ∀t > 0 As it is well known (cf [15, page 482]), any interpolation space X with respect to a -monotone couple (X0, X1) is described by the real -method. Theorem 1.2 allows to find an r.i.s. that contains Rademacher series with coefficients belonging to an arbitrary interpolation space between l1 and l2. In [16, 19], the similar results were obtained by additional conditions with respect to spaces X0 and X1
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