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Fourier coefficients of functions in power-weighted L2-spaces and conditionality constants of bases in Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices Approximations and expansions Harmonic analysis in one variable

Published online by Cambridge University Press:  30 March 2022

J. L. Ansorena Open the ORCID record for J. L. Ansorena [Opens in a new window]
J. L. Ansorena*
Affiliation:
Department of Mathematics and Computer Sciences, Universidad de La Rioja, Logroño 26004, Spain (joseluis.ansorena@unirioja.es)
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Abstract

We prove that, given $2< p<\infty$, the Fourier coefficients of functions in $L_2(\mathbb {T}, |t|^{1-2/p}\,{\rm d}t)$ belong to $\ell _p$, and that, given $1< p<2$, the Fourier series of sequences in $\ell _p$ belong to $L_2(\mathbb {T}, \vert {t}\vert ^{2/p-1}\,{\rm d}t)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1< p<\infty$ and every $0\le \alpha <1$, there is a Schauder basis of $\ell _p$ whose conditionality constants grow as $(m^{\alpha })_{m=1}^{\infty }$, and there is an almost greedy basis of $\ell _p$ whose conditionality constants grow as $((\log m)^{\alpha })_{m=2}^{\infty }$.

Keywords

Fourier coefficientsFourier series, conditionality constants, Hilbert spaces, ℓp-spaces, almost greedy bases

MSC classification

Primary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Secondary: 46B15: Summability and bases 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 3 , June 2023 , pp. 784 - 810
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Albiac, F., Ansorena, J. L., Berasategui, M., Berná, P. M. and Lassalle, S.. Bidemocratic bases and their connections with other greedy-type bases. ArXiv e-prints 2105.15177 (2021).Google Scholar
Albiac, F., Ansorena, J. L., Berná, P. M. and Wojtaszczyk, P.. Greedy approximation for biorthogonal systems in quasi-Banach spaces. Diss. Math. (Rozprawy Mat.) 560 (2021), 1–88.Google Scholar
Albiac, F., Ansorena, J. L., Dilworth, S. J. and Kutzarova, D.. Building highly conditional almost greedy and quasi-greedy bases in Banach spaces. J. Funct. Anal. 276 (2019), 1893–1924.CrossRefGoogle Scholar
Albiac, F., Ansorena, J. L. and Wojtaszczyk, P.. Conditional quasi-greedy bases in non-superre exive Banach spaces. Constr. Approx. 49 (2019), 103–122.10.1007/s00365-017-9399-xCrossRefGoogle Scholar
Albiac, F., Ansorena, J. L. and Wojtaszczyk, P.. On certain subspaces of p for $0 < p \le 1$ and their applications to conditional quasi-greedy bases in p-Banach spaces. Math. Ann. 379 (2021), 465–502.CrossRefGoogle Scholar
Albiac, F. and Kalton, N. J.. Topics in Banach space theory. 2nd edn, Graduate Texts in Mathematics, vol. 233 (Cham: Springer, 2016). With a foreword by Gilles Godefroy.CrossRefGoogle Scholar
Al'tman, M. Š. On bases in Hilbert space. Dokl. Akad. Nauk SSSR (N.S.) 69 (1949), 483–485.Google Scholar
Ansorena, J. L., Bello, G. and Wojtaszczyk, P.. Lorentz spaces and embeddings induced by almost greedy bases in superre exive Banach spaces. ArXiv e-prints 2105.09203 (Accepted for publication in Isr. J. Math., 2021).CrossRefGoogle Scholar
Babenko, K. I.. On conjugate functions. Dokl. Akad. Nauk SSSR (N. S.) 62 (1948), 157–160.Google Scholar
Bergh, J. and Löfström, J.. Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223 (Berlin, New York: Springer-Verlag, 1976).10.1007/978-3-642-66451-9CrossRefGoogle Scholar
Berná, P. M., Blasco, Ó., Garrigós, G., Hernández, E. and Oikhberg, T.. Embeddings and Lebesgue-type inequalities for the greedy algorithm in Banach spaces. Constr. Approx. 48 (2018), 415–451.CrossRefGoogle Scholar
Dilworth, S. J., Kalton, N. J. and Kutzarova, D.. On the existence of almost greedy bases in Banach spaces. Stud. Math. 159 (2003), 67–101. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday.CrossRefGoogle Scholar
Dilworth, S. J., Kalton, N. J., Kutzarova, D. and Temlyakov, V. N.. The thresholding greedy algorithm, greedy bases, and duality. Constr. Approx. 19 (2003), 575–597.10.1007/s00365-002-0525-yCrossRefGoogle Scholar
Dilworth, S. J., Kutzarova, D. and Wojtaszczyk, P.. On approximate $l_{1}$ systems in Banach spaces. J. Approx. Theory 114 (2002), 214–241.10.1006/jath.2001.3641CrossRefGoogle Scholar
Garrigós, G., Hernández, E. and Oikhberg, T.. Lebesgue-type inequalities for quasi-greedy bases. Constr. Approx. 38 (2013), 447–470.CrossRefGoogle Scholar
Garrigós, G. and Wojtaszczyk, P.. Conditional quasi-greedy bases in Hilbert and Banach spaces. Indiana Univ. Math. J. 63 (2014), 1017–1036.10.1512/iumj.2014.63.5269CrossRefGoogle Scholar
Gelbaum, B.. A nonabsolute basis for Hilbert space. Proc. Am. Math. Soc. 2 (1951), 720–721.10.1090/S0002-9939-1951-0043383-6CrossRefGoogle Scholar
Gogyan, S.. An example of an almost greedy basis in $L^{1}(0, 1)$. Proc. Am. Math. Soc. 138 (2010), 1425–1432.CrossRefGoogle Scholar
Gurariĭ, V. I. and Gurariĭ, N. I.. Bases in uniformly convex and uniformly smooth Banach spaces. Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 210–215. English translation in Bases in uniformly convex and uniformly attened Banach spaces. Math. USSR Izv. 220 (1971), 5.Google Scholar
Hunt, R., Muckenhoupt, B. and Wheeden, R.. Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc. 176 (1973), 227–251.10.1090/S0002-9947-1973-0312139-8CrossRefGoogle Scholar
James, R. C.. Super-re exive spaces with bases. Pac. J. Math. 41 (1972), 409–419.10.2140/pjm.1972.41.409CrossRefGoogle Scholar
Konyagin, S. V. and Temlyakov, V. N.. A remark on greedy approximation in Banach spaces. East J. Approx. 5 (1999), 365–379.Google Scholar
Köthe, G. and Toeplitz, O.. Lineare Räume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen. J. Reine Angew. Math. 171 (1934), 193–226.CrossRefGoogle Scholar
Lindenstrauss, J. and Tzafriri, L.. Classical Banach spaces. I – sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas] (Berlin, New York: Springer-Verlag, 1977).10.1007/978-3-642-66557-8_4CrossRefGoogle Scholar
Lindenstrauss, J. and Tzafriri, L.. Classical Banach spaces. II – function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97 (Berlin, New York: Springer-Verlag, 1979).Google Scholar
Pełczyński, A.. Projections in certain Banach spaces. Stud. Math. 19 (1960), 209–228.10.4064/sm-19-2-209-228CrossRefGoogle Scholar
Pełczyński, A. and Singer, I.. On non-equivalent bases and conditional bases in Banach spaces. Stud. Math. 25 (1965), 5–25.10.4064/sm-25-1-5-25CrossRefGoogle Scholar
Wojtaszczyk, P.. Greedy algorithm for general biorthogonal systems. J. Approx. Theory 107 (2000), 293–314.CrossRefGoogle Scholar
Zygmund, A.. Trigonometric series. Vol. I, II, 3rd edn, Cambridge Mathematical Library (Cambridge: Cambridge University Press, 2002). With a foreword by Robert A. Fefferman.Google Scholar