Convergence to infinity for orthonormal spline series
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CONVERGENCE TO INFINITY FOR ORTHONORMAL SPLINE SERIES G. G. GEVORKYAN1,† , K. A. KERYAN1,2,∗,† and M. P. POGHOSYAN1,2 1
2
Faculty of Mathematics and Mechanics, Yerevan State University, Alex Manoogian 1, 0025, Yerevan, Armenia
College of Science and Engineering, American University of Armenia, Marshal Baghramyan 40, 0019, Yerevan, Armenia e-mails: [email protected], [email protected], [email protected] (Received December 19, 2019; accepted March 14, 2020)
Abstract. We generalize an important property of trigonometric series to the case of series by orthonormal spline systems corresponding to the dyadic sequence of grid points. We prove that Ciesielski series cannot diverge to infinity on a set of positive measure.
1. Introduction N. N. Luzin [8] posed the following question: Can a trigonometric series converge to +∞ on a set of positive measure? Since then many mathematicians have studied the question of convergence to +∞ on a set of positive measure for orthogonal series. Yu. B. Germeier proved that a trigonometric series cannot be Riemann summable to +∞ on a set of positive measure. N. N. Luzin and I. I. Privalov [11] constructed a trigonometric series which is almost everywhere Abel summable to +∞. D. E. Men’shov [9] proved that for any measurable function, not necessarily finite almost everywhere, there exists a trigonometric series converging in measure to that function. Particularly there exists a trigonometric series which converges in measure to +∞ on [−π, π]. A. A. Talalyan [14] showed that for any measurable function f on [−π, π] there exists a trigonometric series converging in measure to that function and converging almost everywhere on the set where f is finite. Finally S. V. Konyagin [7] solved the Luzin’s problem by proving the following theorem: ∗ Corresponding
author. first and second authors are supported by SCS RA grant 18T-1A074. Key words and phrases: orthonormal spline system, Ciesielski system, convergence to infinity, martingale property. Mathematics Subject Classification: 42C10, 40A05. † The
c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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G. G. G. G. GEVORKYAN, GEVORKYAN, K. K. A. A. KERYAN KERYAN and and M. M. P. P. POGHOSYAN POGHOSYAN
Theorem 1.1. Let S(x) and S(x) be lower and upper limits of partial sums of a trigonometric series. Then � � mes x ∈ [−π, π] : −∞ < S(x) ≤ S(x) = +∞ = 0.
In particular, a trigonometric series cannot converge to +∞ on a set of positive measure.
Analogous questions are also considered for Haar and Walsh series. A. A. Talalyan and F. G. Arutyunyan [15] proved that Haar and Walsh series cannot converge to +∞ on a set of positive measure. In [6], [13] one can find simpler proofs of that theorem. Nevertheless it was proved in [10] that there exist uniformly bounded orthonormal systems and series with respect to these systems so that for any rearrangement of the series it converges to +∞ on a set of positive measure. N. B. Pogosyan [12] showed that for any complete orthonormal syst
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