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Revista Matemática Complutense

, Volume 32, Issue 1, pp 187–193 | Cite as

The convolution of two differentiable functions on the circle need not be differentiable

  • P. Jiménez-Rodríguez
  • G. A. Muñoz-Fernández
  • E. Sáez-Maestro
  • J. B. Seoane-SepúlvedaEmail author
Article
  • 50 Downloads

Abstract

The convolution operator is well-known for preserving the best properties of its parent functions, and is often presented as a “smoothing” operator. In the present result, we construct two differentiable functions whose convolution is not differentiable.

Keywords

Convolution Non differentiable function Lineability Spaceability Algebrability 

Mathematics Subject Classification

44A35 58B10 54C30 

Notes

Acknowledgements

The authors would like to thank prof. Fedor Nazarov for his kind and selfless help in outlining the ideas that crystallized in the counter examples that resulted in Theorem 2.1.

References

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    Folland, G.B.: Fourier Analysis and Its Applications. The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software. American Mathematical Society, Pacific Grove (1992)zbMATHGoogle Scholar
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    Katznelson, Y.: An Introduction to Harmonic Analysis (3rd Aanalysis). Cambridge Mathematical Library, Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
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    Rudin, W.: Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics, vol. 12. Wiley, New York (1962)zbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  • P. Jiménez-Rodríguez
    • 1
  • G. A. Muñoz-Fernández
    • 2
  • E. Sáez-Maestro
    • 3
  • J. B. Seoane-Sepúlveda
    • 4
    Email author
  1. 1.Department of Mathematical SciencesMathematics and Computer Science BuildingKentUSA
  2. 2.Departamento de Análisis y Matemática Aplicada, Instituto de Matemática Interdisciplinar (IMI), Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  3. 3.Departamento de Análisis y Matemática Aplicada, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  4. 4.Departamento de Análisis Matemático, Instituto de Matemática Interdisciplinar (IMI), Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain

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