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

It is proved that if ${\displaystyle \left\{{\phi }_n\right\}}$ 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 ${\displaystyle \sum _{k=1}^{\infty }{c}_k{\phi }_k\left(x\right)}$, where ${\displaystyle \left\{c_k\right\}\in l_q}$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and ${\displaystyle f\in L_{\mu }^p[0,1]=\{f:{\int }_0^1{\left|f\left(x\right)\right|}^p\mu \left(x\right)dx<\infty \}}$ there are numbers ${\displaystyle {\varepsilon }_k}$, k=1,2,…, ${\displaystyle {\varepsilon }_k}$ = 1 or 0, such that ${\displaystyle \underset{n\to \infty }{lim}{\int }_0^1{|\sum _{k=1}^n{\varepsilon }_k{c}_k{\phi }_k\left(x\right)-f\left(x\right)|}^p\mu \left(x\right)dx=0.}$ 2. For every p ∈ [1,2) and ${\displaystyle f\in L_{\mu }^p[0,1]}$ there are a function ${\displaystyle g\in L^1[0,1]}$ with g(x) = f(x) on E and numbers ${\displaystyle {\delta }_k}$, k=1,2,…, ${\displaystyle {\delta }_k=1}$ or 0, such that ${\displaystyle \underset{n\to \infty }{lim}{\int }_0^1{|\sum _{k=1}^n{\delta }_k{c}_k{\phi }_k\left(x\right)-g\left(x\right)|}^p\mu \left(x\right)dx=0}$, where ${\displaystyle {\delta }_k{c}_k={\int }_0^{1}g\left(t\right){\phi }_k\left(t\right)dt.}$