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On a Property of the Franklin System in C[0, 1] and L1[0, 1]

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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].

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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).

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Authors and Affiliations

  1. Yerevan State University, Yerevan, 375025, Armenia

    V. G. Mikayelyan

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  1. V. G. Mikayelyan
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Correspondence to V. G. Mikayelyan.

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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 2, pp. 241–245.

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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

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  • DOI: https://doi.org/10.1134/S0001434620010289

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