# On the representation of functions by orthogonal series in weighted ${L}^{p}$ spaces

Studia Mathematica (1999)

• Volume: 134, Issue: 3, page 207-216
• ISSN: 0039-3223

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## Abstract

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It is proved that if ${\phi }_{n}$ 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 ${\sum }_{k=1}^{\infty }{c}_{k}{\phi }_{k}\left(x\right)$, where ${c}_{k}\in {l}_{q}$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f\in {L}_{\mu }^{p}\left[0,1\right]=f:{ʃ}_{0}^{1}{|f\left(x\right)|}^{p}\mu \left(x\right)dx<\infty$ there are numbers ${\varepsilon }_{k}$, k=1,2,…, ${\varepsilon }_{k}$ = 1 or 0, such that $li{m}_{n\to \infty }{ʃ}_{0}^{1}{|{\sum }_{k=1}^{n}{\varepsilon }_{k}{c}_{k}{\phi }_{k}\left(x\right)-f\left(x\right)|}^{p}\mu \left(x\right)dx=0.$ 2. For every p ∈ [1,2) and $f\in {L}_{\mu }^{p}\left[0,1\right]$ there are a function $g\in {L}^{1}\left[0,1\right]$ with g(x) = f(x) on E and numbers ${\delta }_{k}$, k=1,2,…, ${\delta }_{k}=1$ or 0, such that $li{m}_{n\to \infty }{ʃ}_{0}^{1}{|{\sum }_{k=1}^{n}{\delta }_{k}{c}_{k}{\phi }_{k}\left(x\right)-g\left(x\right)|}^{p}\mu \left(x\right)dx=0$, where ${\delta }_{k}{c}_{k}={ʃ}_{0}^{1}g\left(t\right){\phi }_{k}\left(t\right)dt.$

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