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On the Uniform Convergence of Spherical Partial Sums of Fourier Series by the Double Walsh System

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Abstract

In this paper, we construct a two-variable integrable function \(U\) whose Fourier coefficients by the double Walsh system are positive on the spectrum and arranged in decreasing order in all directions. For each almost everywhere finite measurable function \(f(x,y)\), \((x,y)\in[0,1)^{2}\), and for any \(\delta>0\) it is possible to find a bounded function \(g(x,y)\) such that

$$|\{(x,y)\in[0,1)^{2}:\ g(x,y)\neq f(x,y)\}|\leq\delta,$$

and \(|c_{k,s}(g)|=c_{k,s}(U)\) on the spectrum of the function \(g\), and its spherical partial sums of the Fourier series by the double Walsh system converge uniformly on \([0,1)^{2}.\)

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REFERENCES

  1. B. Golubov, A. Efimov, and V. Skvortsov, Walsh Series and Transforms, Mathematics and Its Applications, Vol. 64 (Springer, Dordrecht, 1991). https://doi.org/10.1007/978-94-011-3288-6

  2. I. P. Natanson, Theory of Functions of Real Variable (Nauka, Moscow, 1974).

    Google Scholar 

  3. C. Watari, ‘‘Mean convergence of Walsh-Fourier series,’’ Tohoku Math. J. 16, 183–188 (1964). https://doi.org/10.2748/tmj/1178243704

    Article  MathSciNet  MATH  Google Scholar 

  4. M. G. Grigoryan, ‘‘On the universal and strong \((L^{1},L^{\infty})\)-property related to Fourier–Walsh series,’’ Banach J. Math. Anal. 11, 698–712 (2017). https://doi.org/10.1215/17358787-2017-0012

    Article  MathSciNet  MATH  Google Scholar 

  5. M. G. Grigorian, ‘‘Uniform convergence of the greedy algorithm with respect to the Walsh system,’’ Stud. Math. 198 (2), 197–206 (2010). https://doi.org/10.4064/sm198-2-6

    Article  MathSciNet  Google Scholar 

  6. N. N. Luzin, ‘‘On the main theorem of integral calculus,’’ Mat. Sb. 28 (2), 266–294 (1912).

    Google Scholar 

  7. D. E. Men’shov, ‘‘On the uniform convergence of Fourier series,’’ Mat. Sb. 53 (2), 67–96 (1942).

    MATH  Google Scholar 

  8. M. G. Grigoryan and A. A. Sargsyan, ‘‘On the universal function for the class \(L^{p}[0,1],p\in(0,1)\),’’ J. Funct. Anal. 270, 3111–3133 (2016). https://doi.org/10.1016/j.jfa.2016.02.021

    Article  MathSciNet  MATH  Google Scholar 

  9. M. G. Grigoryan and L. N. Galoyan, ‘‘On the universal functions,’’ J. Approximation Theory 225, 191–208 (2018). https://doi.org/10.1016/j.jat.2017.08.003

    Article  MathSciNet  MATH  Google Scholar 

  10. M. G. Grigoryan, ‘‘Universal Fourier series,’’ Math. Notes 108, 282–285 (2020). https://doi.org/10.1134/S0001434620070299

    Article  MathSciNet  MATH  Google Scholar 

  11. M. G. Grigoryan, ‘‘Functions with universal Fourier–Walsh series,’’ Sb.: Math. 211, 850–874 (2020). https://doi.org/10.1070/sm9302

    Article  MathSciNet  MATH  Google Scholar 

  12. M. G. Grigoryan and L. N. Galoyan, ‘‘Functions universal with respect to the trigonometric system,’’ Izv.: Math. 85, 241–261 (2021). https://doi.org/10.1070/im8964

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Kolmogoroff, ‘‘Sur les fonctions harmoniques conjuguées et les séries de Fourier,’’ Fundamenta Math. 7, 24–29 (1925). https://doi.org/10.4064/fm-7-1-24-29

    Article  MATH  Google Scholar 

  14. M. Riss, ‘‘Sur les fonctions conjugees,’’ Math. Z. 27, 214–244 (1927).

    Google Scholar 

  15. L. Carleson, ‘‘On convergence and growth of partial sums of Fourier series,’’ Acta Math. 116, 135–157 (1966). https://doi.org/10.1007/bf02392815

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Fefferman, ‘‘The multiplier problem for the ball,’’ Ann. Math. 94, 330–336 (1971). https://doi.org/10.2307/1970864

    Article  MathSciNet  MATH  Google Scholar 

  17. D. C. Harris, ‘‘Almost everywhere divergence of multiple Walsh–Fourier series,’’ Proc. Am. Math. Soc. 101, 637–643 (1987). https://doi.org/10.1090/s0002-9939-1987-0911024-3

    Article  MathSciNet  MATH  Google Scholar 

  18. M. G. Grigoryan, ‘‘Convergence in the metric of \(L^{p}\), \(o<p<1\), of spherical partial sums of double Fourier series,’’ Math. Notes Acad. Sci. USSR 33, 264–270 (1983). https://doi.org/10.1007/BF01157057

    Article  MATH  Google Scholar 

  19. R. D. Getsadze, ‘‘A continuous function with multiple Fourier series in the Walsh–Paley system that diverges almost everywhere,’’ Math. USSR Sb. 56, 262–286 (1987). https://doi.org/10.1070/SM1987v056n01ABEH003035

    Article  MATH  Google Scholar 

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Funding

The work is supported by the Science Committee of the Republic of Armenia, project no. 21AG-1A066.

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Authors and Affiliations

  1. Yerevan State University, Yerevan, Armenia

    S. A. Sargsyan & L. N. Galoyan

Authors
  1. S. A. Sargsyan
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  2. L. N. Galoyan
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Correspondence to S. A. Sargsyan or L. N. Galoyan.

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The authors declare that they have no conflicts of interest.

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Translated by E. Oborin

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Sargsyan, S.A., Galoyan, L.N. On the Uniform Convergence of Spherical Partial Sums of Fourier Series by the Double Walsh System. J. Contemp. Mathemat. Anal. 58, 370–383 (2023). https://doi.org/10.3103/S1068362323050072

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  • DOI: https://doi.org/10.3103/S1068362323050072

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