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A New Sufficient Condition for Belonging to the Algebra of Absolutely Convergent Fourier Integrals and Its Application to the Problems of Summability of Double Fourier Series

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We establish a general sufficient condition for the possibility of representation of functions

$$ f\left( \max \left\{\left|{x}_1\right|,\left|{x}_2\right|\right\}\right) $$

in the form of absolutely convergent double Fourier integrals and study the possibility of its application to various problems of summability of double Fourier series, in particular, by using the Marcinkiewicz–Riesz method.

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References

  1. 1.

    E. M. Stein and G. Weiss, Introduction to Fourier Analysis of Euclidean Spaces, Princeton Univ. Press, Princeton (1971).

  2. 2.

    E. Liflyand, S. Samko, and R. Trigub, “The Wiener algebra of absolutely convergent Fourier integrals: An overview,” Anal. Math. Phys., 2, No. 1, 1–68 (2012).

  3. 3.

    A. N. Podkorytov, “Summation of multiple Fourier series over polyhedra,” Vestn. Leningrad. Gos. Univ., Mat. Mekh. Astron., No. 1, 51–58 (1980).

  4. 4.

    R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer AP, Dordrecht (2004).

  5. 5.

    E. S. Belinsky, E. R. Liflyand, and R. M. Trigub, “The Banach algebra A∗ and its properties,” Fourier Anal. Appl., 3, No. 2 (1997).

  6. 6.

    G. H. Hardy, Divergent Series, Oxford Univ. Press, Oxford (1949).

  7. 7.

    I. Marcinkiewicz, “Sur une methode remarquable de sommation des series doubles de Fourier,” in: Collected Papers, PWN, Warszawa (1964), pp. 527–538.

  8. 8.

    H. S. Shapiro, “Some Tauberian theorems with applications to approximation theory,” Bull. Amer. Math. Soc., 74, 499–504 (1968).

  9. 9.

    R. M. Trigub, “Linear methods of summation and absolute convergence of Fourier series,” Izv. Akad. Nauk SSSR, Ser. Mat., 32, No. 1, 24–49 (1968).

  10. 10.

    L. V. Zhizhiashvili, “On the summation of double Fourier series,” Sib. Mat. Zh., 8, No. 3, 548–564 (1967).

  11. 11.

    M. F. Timan and V. G. Ponomarenko, “On the approximation of periodic functions of two variables by Marcinkiewicz-type sums,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 9, 59–67 (1975).

  12. 12.

    O. I. Kuznetsova and R. M. Trigub, “Two-sided estimates for the approximation of functions by Riesz and Marcinkiewicz means,” Dokl. Akad. Nauk SSSR, 251, No. 1, 34–36 (1980).

  13. 13.

    R. M. Trigub, “Absolute convergence of Fourier integrals, summability of Fourier series, and approximation of functions on a torus by polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 6, 1378–1409 (1980).

  14. 14.

    B. R. Draganov, “Exact estimates of the rate of approximation of convolution operators,” J. Approxim. Theory, 162, 952–979 (2010).

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Author information

Correspondence to O. V. Kotova.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 8, pp. 1082–1096, August, 2015.

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Kotova, O.V., Trigub, R.M. A New Sufficient Condition for Belonging to the Algebra of Absolutely Convergent Fourier Integrals and Its Application to the Problems of Summability of Double Fourier Series. Ukr Math J 67, 1219–1235 (2016). https://doi.org/10.1007/s11253-016-1147-z

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Keywords

  • Fourier Series
  • Absolute Convergence
  • Arithmetic Means
  • Lebesgue Point
  • Fourier Integral