Rademacher chaos in symmetric spaces. II. (English) Zbl 1084.42508

Astashkin, S. V.

East J. Approx. 6, No. 1, 71-86 (2000).

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.)