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Mat. Sb., 2003, Volume 194, Number 10, Pages 77–106 (Mi msb774)  

This article is cited in 16 scientific papers (total in 16 papers)

On the $L^p_\mu$-strong property of orthonormal systems

M. G. Grigoryan

Yerevan State University

Abstract: Let $\{\varphi_n(x)\}$ be a system of bounded functions complete and orthonormal in $L^2_{[0,1]}$ and assume that $\|\varphi_n\|_{p_0}\leqslant\mathrm{const}$, $n\geqslant 1$, for some $p_0>2$. Then the elements of the system can be rearranged so that the resulting system has the $L^p_\mu$-strong property: for each $\varepsilon>0$ there exists a (measurable) subset $E\subset[0,1]$ of measure $|E|>1-\varepsilon$ and a measurable function $\mu(x)$, $0<\mu(x)\leqslant 1$, $\mu(x)=1$ on $E$ such that for all $p>2$ and $f(x)\in L^p_\mu[0,1]$ one can find a function $g(x)\in L^1_{[0,1]}$ coinciding with $f(x)$ on $E$ such that its Fourier series in the system $\{\varphi_{\sigma(k)}(x)\}$ converges to $g(x)$ in the $L^p_\mu[0,1]$-norm and the sequence of Fourier coefficients of this function belongs to all spaces $l^q$, $q>2$.

DOI: https://doi.org/10.4213/sm774

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English version:
Sbornik: Mathematics, 2003, 194:10, 1503–1532

Bibliographic databases:

UDC: 517.51
MSC: 42C15, 42C20
Received: 24.10.2002

Citation: M. G. Grigoryan, “On the $L^p_\mu$-strong property of orthonormal systems”, Mat. Sb., 194:10 (2003), 77–106; Sb. Math., 194:10 (2003), 1503–1532

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. G. Grigoryan, A. A. Sargsyan, “Non-linear approximation of continuous functions by the Faber-Schauder system”, Sb. Math., 199:5 (2008), 629–653  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. M. G. Grigorian, “On the strengthened $L^1$-greedy property of the Walsh system”, Russian Math. (Iz. VUZ), 52:5 (2008), 20–31  mathnet  crossref  mathscinet  zmath  elib
    3. Grigoryan, MG, “Unconditional C-strong property of Faber-Schauder system”, Journal of Mathematical Analysis and Applications, 352:2 (2009), 718  crossref  mathscinet  zmath  isi  scopus
    4. Grigoryan M., “Uniform convergence of the greedy algorithm with respect to the Walsh system”, Studia Mathematica, 198:2 (2010), 197–206  crossref  mathscinet  zmath  isi  scopus
    5. Grigoryan M.G., Sargsyan A.A., “On the coefficients of the expansion of elements from C[0,1] space by the Faber-Schauder system”, J Funct Spaces Appl, 9:2 (2011), 191–203  crossref  mathscinet  zmath  isi  scopus
    6. M. G. Grigoryan, “Modifications of functions, Fourier coefficients and nonlinear approximation”, Sb. Math., 203:3 (2012), 351–379  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. M. G. Grigoryan, S. A. Sargsyan, “Nonlinear approximation of functions from the class $L^r$ with respect to the Vilenkin system”, Russian Math. (Iz. VUZ), 57:2 (2013), 25–33  mathnet  crossref
    8. Martin Grigoryan, Artavazd Minasyan, “Representation of Functions in L<sup>1</sup><sub style="margin-left:-6px">μ</sub> Weighted Spaces by Series with Monotone Coefficients in the Walsh Genrealized System”, AM, 04:11 (2013), 6  crossref
    9. L. N. Galoyan, M. G. Grigoryan, A. Kh. Kobelyan, “Convergence of Fourier series in classical systems”, Sb. Math., 206:7 (2015), 941–979  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. M. G. Grigoryan, K. A. Navasardyan, “Universal functions in ‘correction’ problems guaranteeing the convergence of Fourier–Walsh series”, Izv. Math., 80:6 (2016), 1057–1083  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. Grigoryan M.G., Sargsyan S., “on the Fourier-Vilenkin Coefficients”, Acta Math. Sci., 37:2 (2017), 293–300  crossref  mathscinet  zmath  isi  scopus
    12. M. G. Grigoryan, A. A. Sargsyan, “The structure of universal functions for $L^p$-spaces, $p\in(0,1)$”, Sb. Math., 209:1 (2018), 35–55  mathnet  crossref  crossref  adsnasa  isi  elib
    13. Grigoryan M.G., Sargsyan S.A., “On the l1-Convergence and Behavior of Coefficients of Fourier-Vilenkin Series”, Positivity, 22:3 (2018), 897–918  crossref  mathscinet  isi  scopus
    14. L. S. Simonyan, “On convergence of the Fourier double series with respect to the Vilenkin systems”, Uch. zapiski EGU, ser. Fizika i Matematika, 52:1 (2018), 12–18  mathnet
    15. M. G. Grigoryan, “On the absolute convergence of Fourier–Haar series in the metric of $L^p(0,1)$, $0<p<1$”, J. Math. Sci. (N. Y.), 243:6 (2019), 844–858  mathnet  crossref
    16. Grigoryan M.G., Sargsyan S.A., “Almost Everywhere Convergence of Greedy Algorithm With Respect to Vilenkin System”, J. Contemp. Math. Anal.-Armen. Aca., 53:6 (2018), 331–345  crossref  mathscinet  isi  scopus
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