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Rademacher chaos in symmetric spaces. II. (English) Zbl 1084.42508
Summary: This paper is a continuation of [S. V. Astashkin, East J. Approx. 4, No. 3, 311–336 (1998; Zbl 1054.46021)] where we started the study of Rademacher chaos in functional symmetric spaces on the segment \([0,1]\).
In this paper we study some properties of the orthonormal system \(\{r_ir_j\}_{1\leq i< j<\infty}\) where \((r_k(t))\) are Rademacher functions on \([0, 1]\), \(k= 1,2,\dots\) . This system is usually called Rademacher chaos of order 2. It is shown that a specific ordering of the chaos leads to a basic sequence (possibly non-unconditional) in a wide class of symmetric functional spaces on \([0, 1]\). Necessary and sufficient conditions on the space are found for the basic sequence \(\{r_i r_j\}_{1\leq i< j<\infty}\) to possess the unconditionality property.

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
41A30 Approximation by other special function classes
42B30 \(H^p\)-spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)