Skip to main content
SpringerLink
Log in

On the order of summability of the Fourier inversion formula

  • Published:
Analysis in Theory and Applications

Abstract

In this article we show that the order of the point value, in the sense of Łojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesàro summable to the distributional point value of order k+1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesàro summable of order k, then the distribution is the (k+1)-th derivative of a locally integrable function, and the distribution has a distributional point value of order k+2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bremermann, H., Distributions, Complex Variables and Fourier Transforms, Addison-Wesley, Reading, Massachusetts, 1965.

    MATH  Google Scholar 

  2. Denjoy, A., Sur l’intégration des coefficients différentiels d’ordre supérieur, Fund. Math., 25(1935), 237–320.

    Google Scholar 

  3. Bochner, S. and Chadrasekharan, K., Fourier Transforms, Annals of Mathematics Studies, No. 19, Princeton University Press, Princeton, N.J., 1949.

    Google Scholar 

  4. Bochner, S., Lectures on Fourier Integrals, Annals of Mathematics Studies, No. 42, Princeton University Press, Princeton, N.J., 1959.

    Google Scholar 

  5. Campos Ferreira, J., Introduction to the theory of distributions, Longman, Harlow, 1997.

    MATH  Google Scholar 

  6. Estrada, R., Characterization of the Fourier Series of a Distribution Having a Value at a point, Proc. Amer. Math. Soc., 124(1996), 1205–1212.

    Article  MATH  MathSciNet  Google Scholar 

  7. Estrada, R., The Cesàro Behaviour of Distributions, Proc. Roy. Soc. London Ser. A, 454(1998), 2425–2443.

    Article  MATH  MathSciNet  Google Scholar 

  8. Estrada, R., A Distributional Version of the Ferenc-Lukács Theorem, Sarajevo J. Math., 1(2005), 1–17.

    MathSciNet  Google Scholar 

  9. Estrada, R. and Kanwal, R. P., A Distributional Approach to Asymptotics. Theory and Applications, second edition, Birkhäuser, Boston, 2002.

    Google Scholar 

  10. Estrada, R. and Vindas, J., Determination of Jumps of Distributions by Differentiated Means, Acta Math. Hungar., 124(2009), 215–241.

    Article  MATH  MathSciNet  Google Scholar 

  11. Fejér, L., Über die Bestimmung des Sprunges der Funktion aus Ihrer Fourierreihe, J. Reine Angew. Math., 142(1913), 165–188.

    Google Scholar 

  12. González Vieli, F. J., Pointwise Fourier Inversion of Distributions, Anal. Theory Appl., 24(2008), 87–92.

    Article  MATH  MathSciNet  Google Scholar 

  13. Gordon, R. A., The integrals of Lebesgue, Denjoy, Perron and Henstock, A.M.S., Providence, 1994.

    MATH  Google Scholar 

  14. Gronwall, T. H., Über eine Summationsmethode und ihre Anwendung auf die Fouriersche Reihe, J. Reine Angew. Math., 147(1916), 16–35.

    Google Scholar 

  15. Hardy, G. H., Divergent Series, Clarendon Press, Oxford, 1949.

    MATH  Google Scholar 

  16. Hardy, G. H. and Littlewood, J. E., Solution of the Cesàro summability problem for power series and Fourier series, Math. Z., 19(1923), 67–96.

    Article  MathSciNet  Google Scholar 

  17. Hardy, G. H. and Littlewood, J. E., The Fourier Series of a Positive Function, J. London Math. Soc., 1(1926), 134–138.

    Article  Google Scholar 

  18. Hardy, G. H., and Riesz, M., The General Theory of Dirichlet’s Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Cambridge University Press, Cambridge, 1952.

    Google Scholar 

  19. Hardy, G. H. and Rogosinski, W. W., Fourier Series, Second Edition, Cambridge Tracts in Mathematics and Mathematical Physics, no. 38, Cambridge, At the University Press, 1950.

    MATH  Google Scholar 

  20. Hobson, E. W., The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Vol. II, Dover Publications, New York, 1958.

    Google Scholar 

  21. Łojasiewicz, S., Sur la Valuer et la Limite D’une Distribution en un Point, Studia Math., 16(1957), 1–36.

    MATH  MathSciNet  Google Scholar 

  22. Lukács, F., Über die Bestimmung des Sprunges Einer Funktion aus Ihrer Fourierreihe, J. Reine Angew. Math., 150(1920), 107–112.

    Google Scholar 

  23. Marcinkiewicz, J., Sur les séries de Fourier, Fundamenta Mathematicae, 27(1936), 38–69.

    Google Scholar 

  24. Móricz, F., Determination of Jumps in Terms of Abel-Poisson Means, Acta. Math. Hungar., 98(2003), 259–262.

    Article  MATH  MathSciNet  Google Scholar 

  25. Móricz, F., Ferenc Lukács type Theorems in Terms of the Abel-Poisson Means of Conjugate Series, Proc. Amer. Math. Soc., 131(2003), 1243–1250.

    Article  MATH  MathSciNet  Google Scholar 

  26. Pilipović, S., Stanković, B. and Takači, A., Asymptotic Behaviour and Stieltjes Transformation of Distributions, Teubner-Texte zur Mathematik, Leipzig, 1990.

    MATH  Google Scholar 

  27. Plancherel, M., Sur la Convergence et sur la Sommation par les Moyennes de Cesàro de limz→∞∫ za f(x) cos xydx, Math. Annalen, 76(1915), 315–326.

    Article  MathSciNet  Google Scholar 

  28. Riesz, M., Une méthode de Sommation équivalente à la méthode des Moyennes Arithmétiques, C. R. Acad. Sci. Paris, (1911), 1651–1654.

  29. Schwartz, L., Théorie des Distributions, Hermann, Paris, 1966.

    MATH  Google Scholar 

  30. Seneta, E., Regularly Varying Functions, Lecture Notes in Mathematics, 598, Springer Verlag, Berlin, 1976.

    MATH  Google Scholar 

  31. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, Second Edition, Clarendon Press, Oxford, 1948.

    Google Scholar 

  32. de la Vallée Poussin, Ch. J., Sur L’approximation des Fonctions D’une Variable réelle et Leurs dérivées par les polynômes et les Suites limitées de Fourier, Bull. de l’Acd. Royale de Belgique, (1908), 193–254.

  33. Vindas, J., Structural Theorems for Quasiasymptotics of Distributions at Infinity, Pub. Inst. Math. (Beograd) (N.S.), 84(2008), 159–174.

    MathSciNet  Google Scholar 

  34. Vindas, J., The Structure of Quasiasymptotics of Schwartz Distributions, Linear and non-linear Theory of Generalized Functions and its Applications, Banach Center Publ. 88, Polish Acad. Sc. Inst. Math., Warsaw, 2010.

    Google Scholar 

  35. Vindas, J. and Estrada, R., Distributionally Regulated Functions, Studia. Math., 181(2007), 211–236.

    Article  MATH  MathSciNet  Google Scholar 

  36. Vindas, J. and Estrada, R., Distributional Point Values and Convergence of Fourier Series and Integrals, J. Fourier Anal. Appl., 13(2007), 551–576.

    Article  MATH  MathSciNet  Google Scholar 

  37. Vindas, J. and Estrada, R., On the Jump Behavior and Logarithmic Averages, J. Math. Anal. Appl., 347(2008), 597–606.

    Article  MATH  MathSciNet  Google Scholar 

  38. Vindas, J. and Estrada, R., On the Support of Tempered Distributions, Proc. Edinb. Math. Soc., 53:1(2010), 255–270.

    Article  MATH  Google Scholar 

  39. Vindas, J. and Pilipović, S., Structural Theorems for Quasiasymptotics of Distributions at the Origin, Math. Nachr., 282(2009), 1584–1599.

    Article  MATH  MathSciNet  Google Scholar 

  40. Vladimirov, V. S., Drozhzhinov, Yu. N. and Zavialov, B. I., Tauberian Theorems for Generalized Functions, Kluwer Academic Publishers, Dordrecht, 1988.

    MATH  Google Scholar 

  41. Vladimirov, V. S., Methods of the Theory of Generalized Functions, Taylor & Francis, London, 2002.

    MATH  Google Scholar 

  42. Walter, G., Pointwise Convergence of Distribution Expansions, Studia Math., 26(1966), 143–154.

    MATH  MathSciNet  Google Scholar 

  43. Walter, G., Fourier Series and Analytic Representation of Distributions, SIAM Review, 12(1970), 272–276.

    Article  MATH  MathSciNet  Google Scholar 

  44. Walter, G. and Shen, X.,Wavelets and other Orthogonal Systems, Second Edition, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, 2001.

    MATH  Google Scholar 

  45. Zygmund, A., Sur un théorème de M. Gronwall, Bull. Acad. Polon., (1925), 207–217.

  46. Zygmund, A., Trigonometric Series, Vols. I & II, Second Edition, Cambridge University Press, New York, 1959.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics & Computer Algebra, Ghent University, Krijgslaan 281 Building S22, B9000, Gent, Belgium

    Jasson Vindas

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Ricardo Estrada

Authors
  1. Jasson Vindas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ricardo Estrada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jasson Vindas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vindas, J., Estrada, R. On the order of summability of the Fourier inversion formula. Anal. Theory Appl. 26, 13–42 (2010). https://doi.org/10.1007/s10496-010-0013-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10496-010-0013-3

Key words

AMS (2010) subject classification

Search

Navigation