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Convergence to infinity for orthonormal spline series

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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.

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Reference

  1. Böhm, W.: Inserting new knots into B-spline curves. Computer-Aided Design 12, 199–201 (1980)

    Article  Google Scholar 

  2. Gevorkyan, G.G.: On the uniqueness of series in the Franklin system. Sb. Math. 207, 1650–1673 (2016)

    Article  MathSciNet  Google Scholar 

  3. Gevorkyan, G.G.: Uniqueness theorem for multiple Franklin series. Math. Notes 101, 219–229 (2017)

    Article  MathSciNet  Google Scholar 

  4. Gevorkyan, G.G.: Uniqueness theorems for Franklin series converging to integrable functions. Sb. Math. 209, 802–822 (2018)

    Article  MathSciNet  Google Scholar 

  5. Gevorkyan, G.G.: On convergence of Franklin series to \(+\infty \). Math. Notes 106, 334–341 (2019)

    Article  MathSciNet  Google Scholar 

  6. Gundy, R.F.: Martingale theory and pointwise convergence of certain orthogonal series. Trans. Amer. Math. Soc. 124, 228–248 (1966)

    Article  MathSciNet  Google Scholar 

  7. Konyagin, S.V.: Limits of indeterminacy of trigonometric series. Math. Notes 44, 910–920 (1988)

    Article  MathSciNet  Google Scholar 

  8. N. N. Luzin, The Integral and Trigonometric Series, GITTL (1951) (in Russian)

  9. D. E. Men′shov, On convergence in measure of trigonometric series, Trudy Mat. Inst. Steklov, 32, : (Russian); translation in Amer. Math. Soc. Transl. 3(1962), 196–270 (1950)

  10. Ovsepyan, R.I., Talalyan, A.A.: Convergence of orthogonal series to \(+\infty \). Math. Notes 8, 545–549 (1970)

    Article  Google Scholar 

  11. I. I. Privalov, Boundary Properties of Analytic Functions, SPTTL (1950) (in Russian)

  12. Pogosyan, N.B.: Representation of measurable functions by bases of \(L^p[0,1]\), \(p\ge 2\). Dokl. Akad. Nauk Arm SSR 63, 205–209 (1976)

    Google Scholar 

  13. Skvortsov, V.A.: Differentiation with respect to nets and the Haar series. Math. Notes 4, 509–513 (1968)

    Article  MathSciNet  Google Scholar 

  14. Talalyan, A.A.: Trigonometric series universal with respect to its subseries. Izv. Akad. Nauk SSSR Ser. Mat. 27, 621–660 (1963)

    MathSciNet  Google Scholar 

  15. A. A. Talalyan and F. G. Arutyunyan, On the convergence to \(+\infty \), Mat. Sb. (N.S.), 66(108) (1965), 240–247

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Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Yerevan State University, Alex Manoogian 1, 0025, Yerevan, Armenia

    G. G. Gevorkyan, K. A. Keryan & M. P. Poghosyan

  2. College of Science and Engineering, American University of Armenia, Marshal Baghramyan 40, 0019, Yerevan, Armenia

    K. A. Keryan & M. P. Poghosyan

Authors
  1. G. G. Gevorkyan
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  2. K. A. Keryan
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  3. M. P. Poghosyan
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Corresponding author

Correspondence to K. A. Keryan.

Additional information

The first and second authors are supported by SCS RA grant 18T-1A074.

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Gevorkyan, G.G., Keryan, K.A. & Poghosyan, M.P. Convergence to infinity for orthonormal spline series. Acta Math. Hungar. 162, 604–617 (2020). https://doi.org/10.1007/s10474-020-01051-4

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  • DOI: https://doi.org/10.1007/s10474-020-01051-4

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