Armenian Journal of Mathematics

Current Issue Vol. 18 No. 2 (2026): On Weyl Multipliers of Almost Everywhere Unconditional Convergence of Series With Respect to General Haar and Franklin Systems

• Articles

  On Weyl Multipliers of Almost Everywhere Unconditional Convergence of Series With Respect to General Haar and Franklin Systems

  Rafayel Barkhudaryan, Gegham Gevorkyan

  View PDF Article Preview

  Abstract

  We obtain necessary and sufficient conditions for an increasing nonnegative sequence $\omega_n$, $n\in\mathbb{N}$, to be a Weyl multiplier for a.e. unconditional convergence for general Haar or Franklin systems corresponding to an admissible sequence ${\cal T}$.

  References

  S.V. Bočkarev, Rearrangements of Fourier-Walsh series, Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), no. 5, pp. 1025-1041.

  Z. Ciesielski, Constructive function theory and spline systems, Studia Math., 53 (1975), no. 3, pp. 277-302.

  Z. Ciesielski and A. Kamont, Projections onto piecewise linear functions, Funct. Approx. Comment. Math., 25 (1997), pp. 129-143.

  P. Franklin, A set of continuous orthogonal functions, Math. Ann., 100 (1928), no. 1, pp. 522-529.

  G.G. Gevorkyan, On Weyl factors for the unconditional convergence of series in the Franklin system, Mat. Zametki, 41 (1987), no. 6, pp. 789-797.

  G.G. Gevorkyan, Absolute and unconditional convergence of series in the Franklin system, Mat. Zametki, 45 (1989), no. 3, pp. 30-42.

  G.G. Gevorkyan, On Weyl multipliers for unconditional convergence of series in Ciesielski systems, Mat. Zametki, 116 (2024), no. 5-6, pp. 707-713.

  G. Gevorkyan, On the Weyl multipliers for general Haar and Franklin systems, J. Contemp. Math. Anal., 59 (2024), no. 4, pp. 262-271.

  G.G. Gevorkyan, Weyl UC-multipliers for Strömberg wavelets, Sib. Math. J., 66 (2025), no. 2, pp. 273-278.

  G. Gevorkyan and A. Kamont, On general Franklin systems, Dissertationes Math. (Rozprawy Mat.), 374, 59 pp., 1998.

  G.G. Gevorkyan and K.A. Keryan, On the absolute and unconditional convergence of series in the general Franklin system, Izv. Ross. Akad. Nauk Ser. Mat., 73 (2009), no. 2, pp. 69-90.

  A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann., 69 (1910), no. 3, pp. 331-371.

  A. Kamont and G.A. Karagulyan, On wavelet polynomials and Weyl multipliers, J. Anal. Math., 150 (2023), no. 2, pp. 529-545.

  G. A. Karagulyan, On Weyl multipliers of a rearranged trigonometric system, Mat. Sb., 211 (2020), no. 12, pp. 49-82.

  G.A. Karagulyan, On uc-multipliers for multiple trigonometric systems, preprint arXiv:2601.10360, 2026.

  B.S. Kashin and A.A. Saakyan, Ortogonolnye ryady. Izdatelstvo Nauchno-Issledovatelskogo Aktuarno-Finansovogo Tsentra (AFTs), Moscow, second ed., 1999.

  S. Kačmaž and G. Šteĭngauz, Teoriya ortogonalnykh ryadov, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958.

  S. Nakata, On the unconditional convergence of Walsh series, Anal. Math., 5 (1979), no. 3, pp. 201-205.

  E.M. Nikišin and P.L. Uljanov, On absolute and unconditional convergence, Uspehi Mat. Nauk, 22 (1967), no. 3(135), pp.240-242.

  J.O. Strömberg, A modified Franklin system and higher-order spline systems on $R^n$ as unconditional bases for Hardy spaces, in Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., pp. 475-494, Wadsworth, Belmont, CA, 1983.

  P.L. Uljanov, Divergent Fourier series, Uspehi Mat. Nauk, 16, (1961), no. 3(99), pp.61-142.

  P.L. Ul'janov, Weyl factors for unconditional convergence, Mat. Sb. (N.S.), 60 (1963), no. 102, pp. 39-62