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Examples of divergent fourier series for classes of functions of bounded Λ-variation

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Abstract

The author has shown earlier that the requirement that a continuous function belong to the class HBV ([−π, π]m) for m ≥ 3 is not sufficient for the convergence of its Fourier series over rectangles. The author gave examples of functions of three and more variables from the Waterman class which are harmonic in the first variable and significantly narrower in the other variables and whose Fourier series are divergent at some point even on cubes. In the present paper, this assertion is strengthened. The main result is that such an example can be constructed even when the class with respect to the first variable is somewhat narrowed. Also the one-dimensional result due to Waterman is refined.

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References

  1. D. Waterman, “On convergence of Fourier series of functions of generalized bounded variation,” Studia Math. 44(1), 107–117 (1972).

    MATH  MathSciNet  Google Scholar 

  2. A. A. Saakyan, “On the convergence of double Fourier series of functions with bounded harmonic variation,” Izv. Akad. Nauk Arm. SSR, Mat. 21(6), 517–529 (1986) [Sov. J. Contemp.Math. Anal., Arm. Acad. Sci. 21 (6), 1–13 (1986)].

    MATH  MathSciNet  Google Scholar 

  3. A. I. Sablin, Functions of Bounded Λ-Variation and Fourier Series, Candidate Dissertation in Mathematics and Physics (Moscow State University, Moscow, 1987) [in Russian].

    Google Scholar 

  4. A. I. Sablin, “Λ-variation and Fourier series,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 10, 66–68 (1987).

  5. A. N. Bakhvalov, “On convergence and localization of multiple Fourier series for classes of functions of bounded Λ-variation,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 3, 6–12 (2008).

  6. A. N. Bakhvalov, “Continuity in Λ-variation of functions of several variables and the convergence of multiple Fourier series. (English. Russian original) [J] Sb. Math. 193, No. 12, 1731–1746 (2002 Mat. Sb. 193 (12), 3–20 (2002) [Russian Acad. Sci. Sb.Math., No. 12, 1731–1746 (2002)].

  7. A. N. Bakhvalov, “Representation of nonperiodic functions of bounded Λ-variation by the Fourier integral in the multidimensional case,” Izv. Ross. Akad. Nauk Ser. Mat. 67(6), 3–22 (2003) [Russian Acad. Sci. Izv. Math. 67 (6), 1081–1100 (2003)].

    MathSciNet  Google Scholar 

  8. S. Perlman and D. Waterman, “Some remarks on functions of Λ-bounded variation,” Proc. Amer. Math. Soc. 74(1), 113–118 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  9. N. K. Bari, Trigonometric Series (Fizmatlit, Moscow, 1961) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    A. N. Bakhvalov

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  1. A. N. Bakhvalov
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Correspondence to A. N. Bakhvalov.

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Original Russian Text © A. N. Bakhvalov, 2009, published in Matematicheskie Zametki, 2009, Vol. 86, No. 5, pp. 664–672.

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Bakhvalov, A.N. Examples of divergent fourier series for classes of functions of bounded Λ-variation. Math Notes 86, 629–636 (2009). https://doi.org/10.1134/S0001434609110042

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