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

Grigorian, M.

On the representation of functions by orthogonal series in weighted

L^{p}

spaces

Grigorian, M.

Studia Mathematica, Tome 133 (1999), p. 207-216 / Harvested from The Polish Digital Mathematics Library

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Résumé

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 $\sum_{k = 1}^{\infty} c_{k} φ_{k} (x)$ , where $c_{k} \in l_{q}$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f \in L_{μ}^{p} [0, 1] = f : ʃ_{0}^{1} {| f (x) |}^{p} μ (x) d x < \infty$ there are numbers $ɛ_{k}$ , k=1,2,…, $ɛ_{k}$ = 1 or 0, such that $l i m_{n \to \infty} ʃ_{0}^{1} {| \sum_{k = 1}^{n} ɛ_{k} c_{k} φ_{k} (x) - f (x) |}^{p} μ (x) d x = 0 .$ 2. For every p ∈ [1,2) and $f \in L_{μ}^{p} [0, 1]$ there are a function $g \in L^{1} [0, 1]$ with g(x) = f(x) on E and numbers $δ_{k}$ , k=1,2,…, $δ_{k} = 1$ or 0, such that $l i m_{n \to \infty} ʃ_{0}^{1} {| \sum_{k = 1}^{n} δ_{k} c_{k} φ_{k} (x) - g (x) |}^{p} μ (x) d x = 0$ , where $δ_{k} c_{k} = ʃ_{0}^{1} g (t) φ_{k} (t) d t .$

Publié le : 1999-01-01

Zbl 0917.42030

EUDML-ID : urn:eudml:doc:216634

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     title = {On the representation of functions by orthogonal series in weighted $L^p$ spaces},
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     year = {1999},
     pages = {207-216},
     zbl = {0917.42030},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv134i3p207bwm}
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Grigorian, M. On the representation of functions by orthogonal series in weighted $L^p$ spaces. Studia Mathematica, Tome 133 (1999) pp. 207-216. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv134i3p207bwm/

Bibliographie

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