On the representation of functions by orthogonal series in weighted $L^p$ spaces
Volume 134 / 1999
Abstract
It is proved that if ${φ_n}$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form $∑^{∞}_{k=1} c_{k}φ_{k}(x)$, where ${c_k} ∈ l_q$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f ∈ L^{p}_{μ}[0,1] = {f:ʃ^{1}_{0}|f(x)|^{p} μ(x)dx < ∞}$ there are numbers $ɛ_k$, k=1,2,…, $ɛ_k$ = 1 or 0, such that $lim_{n→∞} ʃ^{1}_{0}|∑^n_{k=1}ɛ_{k}c_{k}φ_{k}(x)-f(x)|^{p} μ(x)dx = 0.$ 2. For every p ∈ [1,2) and $f ∈ L^p_μ[0,1]$ there are a function $g ∈ L^1[0,1]$ with g(x) = f(x) on E and numbers $δ_{k}$, k=1,2,…, $δ_{k}=1$ or 0, such that $lim_{n→∞} ʃ^{1}_{0}|∑^n_{k=1}δ_{k}c_{k}φ_{k}(x) - g(x)|^{p} μ(x)dx=0$, where $δ_{k}c_{k}=ʃ^1_0g(t)φ_{k}(t)dt.$
