Abstract
In this paper, we prove that if {nk} is an arbitrary increasing sequence of natural numbers such that the ratio nk+1/nk is bounded, then the nk-th partial sum of a series by Franklin system cannot converge to +∞ on a set of positive measure. Also, we prove that if the ratio nk+1/nk is unbounded, then there exists a series by Franklin system, the nk-th partial sum of which converges to +∞ almost everywhere on [0, 1].
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References
N. N. Lusin, Integral and Trigonometric Series (GITTL, Moscow-Leningrad, 1951).
Yu. B. Germeier, Derivatives of Riemann and Valiée Poussin and their application to problems from the theory of trigonometric series (Candidate’s Dissertation, Phys-Math. Sciences, Moscow, 1946).
I. I. Privalov, Boundary Properties of Analytic Functions (GITTL, Moscow, 1950).
D. E. Men’shov, “On convergence in measure of trigonometric series”, Trudy MIAN SSSR, 32, 3–97, 1950.
A. A. Talalyan, “Trigonometric series which are universal with respect to subseries”, Izv. Akad. Nauk SSSR, Ser. Mat., 27(3), 621–660, 1963.
S. V. Konyagin, “Limits of indeterminacy of trigonometric series”, Mat. Zametki, 44(6), 770–784, 1988.
A. A. Talalyan, F. G. Artuyunyan, “On convergence of a series in a Haar system to ∞”, Mat. Sb., 66(2), 240–247, 1965.
R. F. Gundy, “Martingale theory and pointwise convergence of certain orthogonal series”, Trans. Amer. Math. Soc., 124(2), 228–248, 1966.
V. A. Skvortsov, “Differentiation with respect to nets and Haar series”, Mat. Zametki, 4(1), 33–40, 1968.
R.I. Ovsepyan, A.A. Talalyan, “On convergence of orthogonal series to ∞”, Mat. Zametki, 8(2), 129–135, 1970.
N. B. Pogosyan, “Representation of measurable functions by bases in L p[0, 1], p > 2”, DAN Arm. SSR, 63(4), 205–209, 1976.
G. G. Gevorkyan, “On convergence of Franklin series to +∞”, Math. Notes, 106(3), 334Ц341, 2019.
Ph. Franklin, “A set of continuous orthogonal functions”, Math. Annalen, 100, 522–529, 1928.
G. G. Gevorkyan, “On the uniqueness of series in the Franklin system”, Mat. Sb., 207(12), 30–53, 2016.
G. G. Gevorkyan, “Uniqueness theorems for Franklin series converging to integrable functions”, Mat. Sb., 209(6), 25–46, 2018.
G. G. Gevorkyan, “Uniqueness theorem for multiple Franklin series”, Mat. Zametki, 101(2), 199–210, 2017.
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Russian Text © The Author(s), 2019, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2019, No. 6, pp. 54–65.
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Navasardyan, K.A., Mikayelyan, V.G. On Convergence of Partial Sums of Franklin Series to +∞. J. Contemp. Mathemat. Anal. 54, 347–354 (2019). https://doi.org/10.3103/S1068362319060049
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DOI: https://doi.org/10.3103/S1068362319060049