Series by Haar System with Monotone Coefficients

摘要

Let X = {Xn}_(n=0)~∞ be the Haar system normalized in L~2(0.1) and M = {M_x}_(s=1)~∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of X of the form (φk} = Xs = {Xn}n∈S, where S = S(M) = {n_k}_(k=1)~∞ = {n ∈ V[p] : p ∈ M}, V[0] = {1,2} and V[p] = {2~p + 1, 2~p + 2,…, 2~(p+1)} for p = 1, 2,… a series of the form ∑_(I=1)~∞ a_iφ_I with a_I ↘ 0 is constructed, that is universal with respect to partial series in all classes L~r(0,1), r ∈ (0,1), in the sense of a.e. convergence and in the metric of L~r(0,1). The constructed series is universal in the class of all measurable, finite functions on [0,1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ε > 0 there exists a measurable function μ(x), x ∈ [0,1], such that 0 ≤μ(x) ≤ 1 and |{x ∈ [0,1], μ(x) ≠ 1}| < ε, and the series is universal in the weighted space L_μ[0,1] with respect to subseries, in the sense of convergence in the norm of L_μ[0,1].