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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.

##### MSC:
 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.)