# 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

top

## Abstract

top
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 ${\delta }_{k}{c}_{k}={ʃ}_{0}^{1}g\left(t\right){\phi }_{k}\left(t\right)dt.$

## How to cite

top

Grigorian, M.. "On the representation of functions by orthogonal series in weighted $L^p$ spaces." Studia Mathematica 134.3 (1999): 207-216. <http://eudml.org/doc/216634>.

@article{Grigorian1999,
author = {Grigorian, M.},
journal = {Studia Mathematica},
keywords = {representation of functions by orthogonal series; weighted spaces; complete orthonormal system},
language = {eng},
number = {3},
pages = {207-216},
title = {On the representation of functions by orthogonal series in weighted $L^p$ spaces},
url = {http://eudml.org/doc/216634},
volume = {134},
year = {1999},
}

TY - JOUR
AU - Grigorian, M.
TI - On the representation of functions by orthogonal series in weighted $L^p$ spaces
JO - Studia Mathematica
PY - 1999
VL - 134
IS - 3
SP - 207
EP - 216
LA - eng
KW - representation of functions by orthogonal series; weighted spaces; complete orthonormal system
UR - http://eudml.org/doc/216634
ER -

## References

top
1. [1] N. K. Bari, Trigonometric Series, Fizmatgiz, Moscow, 1961 (in Russian).
2. [2] M. G. Grigorian, On convergence of Fourier series in complete orthonormal systems in the ${L}^{1}$ metric and almost everywhere, Mat. Sb. 181 (1990), 1011-1030 (in Russian); English transl.: Math. USSR-Sb. 70 (1991), 445-466.
3. [3] M. G. Grigorian, On the convergence of Fourier series in the metric of ${L}^{1}$, Anal. Math. 17 (1991), 211-237. Zbl0754.42006
4. [4] M. G. Grigorian, On some properties of orthogonal systems, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 5, 75-105.
5. [5] N. N. Luzin, On the fundamental theorem of the integral calculus, Mat. Sb. 28 (1912), 266-294 (in Russian).
6. [6] D. E. Men'shov, On Fourier series of integrable functions, Trudy Moskov. Mat. Obshch. 1 (1952), 5-38.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.