Abstract
A problem posed by J. R. Holub is solved. In particular, it is proved that if \(\left\{ {{{\tilde f}_n}} \right\}\) is the normalized Franklin system in L1[0, 1], {an} is a monotone sequence converging to zero, and \({\sup\nolimits _{n \in \mathbb{N} }}{\left\| {\sum\nolimits_{k = 0}^n {{a_k}{{\tilde f}_k}} } \right\|_1}\, < \, + \infty \), then the series \(\sum\nolimits_{n = 0}^\infty {{a_n}{{\tilde f}_n}} \) converges in L1[0, 1]. A similar result is also obtained for C[0, 1].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. M. Day, Normed Linear Spaces (Springer-Verlag, Berlin, 1962).
J. R. Holub, “Bounded completeness and Schauder’s basis for C[0, 1],” Glasgow Math. J. 28 (1), 15–19 (1986).
V. Kadets, “The Haar system in L1 is monotonically boundedly complete,” Mat. Fiz. Anal. Geom. 12 (1), 103–106 (2005).
Ph. Franklin, “A set of continuous orthogonal functions,” Math. Ann. 100 (1), 522–529 (1928).
S. V. Bočkarev, “Some inequalities for the Franklin series,” Anal. Math. 1 (4), 249–257 (1975).
Z. Ciesielski, “Properties of the orthonormal Franklin system. II,” Studia Math. 27 (3), 289–323 (1966).
P. F. X. Müller and M. Passenbrunner, Almost Everywhere Convergence of Spline Sequences, arXiv: 1711.01859 (2019).
G. G. Gevorkyan, “On series in the Franklin system,” Anal. Math. 16 (2), 87–114 (1990).
S. V. Bochkarev, “Existence of a basis in the space of functions analytic in the disk, and some properties of Franklin’s system,” Mat. Sb. 95 (137) (1 (9)), 3–18 (1974).
S. V. Bochkarev, Math. USSR-Sb. 24 (1), 1–16 (1974).
G. G. Gevorkyan, “Unboundedness of the shift operator with respect to the Franklin system in the space L1” Mat. Zametki 38 (4), 523–533 (1985).
G. G. Gevorkyan, Math. Notes 38 (4), 796–802 (1985).
Acknowledgments
The author wishes to express gratitude to Academician G. G. Gevorkyan for his advice during the work on the present paper.
Funding
This work was supported by the State Committee for Science of the Ministry of Education and Science of the Republic of Armenia (project GKN RA 10-3/1-41).
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 2, pp. 241–245.
Rights and permissions
About this article
Cite this article
Mikayelyan, V.G. On a Property of the Franklin System in C[0, 1] and L1[0, 1]. Math Notes 107, 284–287 (2020). https://doi.org/10.1134/S0001434620010289
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620010289