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Asymptotic Behavior of the Fourier Transform of a Function of Bounded Variation

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Book cover Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

A recent result on the asymptotic behavior of the sine Fourier transform of an arbitrary locally absolutely continuous function of bounded variation is extended to the case of several variables. For this, the initial one-dimensional result is reconsidered and refined. To even one-dimensional asymptotics and their multidimensional generalizations, a new balance operator is introduced.

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References

  1. C.R. Adams, J.A. Clarkson, Properties of functions f(x, y) of bounded variation. Trans. Am. Math. Soc. 36, 711–730 (1934)

    Google Scholar 

  2. H. Bateman, A. Erdélyi, Tables of Integral Transforms, vol. II (McGraw Hill Book Company, New York, 1954)

    Google Scholar 

  3. E. Berkson, T.A. Gillespie, Absolutely continuous functions of two variables and well-bounded operators. J. Lond. Math. Soc. (2) 30, 305–324 (1984)

    Google Scholar 

  4. D. Borwein, Linear functionals connected with strong Cesáro summability. J. Lond. Math. Soc. 40, 628–634 (1965)

    Article  MATH  Google Scholar 

  5. J.A. Clarkson, C.R. Adams, On definitions of bounded variation for functions of two variables. Trans. Am. Math. Soc. 35, 824–854 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Fridli, Hardy spaces generated by an integrability condition. J. Approx. Theory 113, 91–109 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Ganzburg, E. Liflyand, Estimates of best approximation and fourier transforms in integral metrics. J. Approx. Theory 83, 347–370 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics (North-Holland, Amsterdam, 1985)

    MATH  Google Scholar 

  9. D.V. Giang, F. Móricz, Lebesgue integrability of double Fourier transforms. Acta Sci. Math. (Szeged) 58, 299–328 (1993)

    MathSciNet  MATH  Google Scholar 

  10. G.H. Hardy, On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Q. J. Math. 37, 53–79 (1906)

    MATH  Google Scholar 

  11. E.W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, vol. 1, 3rd edn. (University Press, Cambridge, 1927; Dover, New York, 1957)

    Google Scholar 

  12. A. Iosevich, E. Liflyand, Decay of the Fourier Transform: Analytic and Geometric Aspects (Birkhäuser, Heidelberg, 2014)

    Book  MATH  Google Scholar 

  13. H. Kober, A note on Hilbert’s operator. Bull. Am. Math. Soc. 48(1), 421–426 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Lerner, E. Liflyand, Interpolation properties of a scale of spaces. Collect. Math. 54, 153–161 (2003)

    MathSciNet  MATH  Google Scholar 

  15. E. Liflyand, Fourier transforms of functions from certain classes. Anal. Math. 19, 151–168 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Liflyand, Fourier transform versus Hilbert transform. Ukr. Math. Bull. 9, 209–218 (2012); Also published in J. Math. Sci. 187, 49–56 (2012)

    Google Scholar 

  17. E. Liflyand, Integrability spaces for the Fourier transform of a function of bounded variation. J. Math. Anal. Appl. 436, 1082–1101 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. E. Liflyand, Multiple Fourier transforms and trigonometric series in line with Hardy’s variation. Contemp. Math. 659, 135–155 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Liflyand, S. Tikhonov, Weighted Paley-Wiener theorem on the Hilbert transform. C.R. Acad. Sci. Paris Ser. I 348, 1253–1258 (2010)

    Google Scholar 

  20. E. Liflyand, S. Samko, R. Trigub, The wiener algebra of absolutely convergent Fourier integrals: an overview. Anal. Math. Phys. 2, 1–68 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. A.A. Talalyan, G.G. Gevorkyan, Representation of absolutely continuous functions of several variables. Acta Sci. Math. (Szeged) 54, 277–283 (1990) (Russian)

    MathSciNet  MATH  Google Scholar 

  22. R.M. Trigub, On integral norms of polynomials. Matem. Sbornik 101(143), 315–333 (1976) (Russian); English transl. in Math. USSR Sbornik 30, 279–295 (1976)

    Google Scholar 

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Acknowledgements

The author is grateful to the referee for thorough reading and numerous useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Elijah Liflyand

Authors
  1. Elijah Liflyand
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Correspondence to Elijah Liflyand .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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Liflyand, E. (2017). Asymptotic Behavior of the Fourier Transform of a Function of Bounded Variation. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55556-0_4

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