PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Studia Mathematica

1999 | 134 | 3 | 207-216
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
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 $δ_{k}c_{k}=ʃ^1_0g(t)φ_{k}(t)dt.$
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
207-216
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-02-07
poprawiono
1998-02-17
Twórcy
autor
• Department of Physics, State University of Yerevan, Alek Manukian 1, 375019 Yerevan, Republic of Armenia
Bibliografia
• [1] N. K. Bari, Trigonometric Series, Fizmatgiz, Moscow, 1961 (in Russian).
• [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] M. G. Grigorian, On the convergence of Fourier series in the metric of $L^1$, Anal. Math. 17 (1991), 211-237.
• [4] M. G. Grigorian, On some properties of orthogonal systems, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 5, 75-105.
• [5] N. N. Luzin, On the fundamental theorem of the integral calculus, Mat. Sb. 28 (1912), 266-294 (in Russian).
• [6] D. E. Men'shov, On Fourier series of integrable functions, Trudy Moskov. Mat. Obshch. 1 (1952), 5-38.
Typ dokumentu
Bibliografia
Identyfikatory