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.
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References
Ferrera, J.: An Introduction to Nonsmooth Analysis. Elsevier, Oxford (2014)
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)
Katznelson, Y.: An Introduction to Harmonic Analysis (3rd Aanalysis). Cambridge Mathematical Library, Cambridge University Press, Cambridge (2004)
Rudin, W.: Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics, vol. 12. Wiley, New York (1962)
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.
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Supported by Grant MTM2015-65825-P.
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Jiménez-Rodríguez, P., Muñoz-Fernández, G.A., Sáez-Maestro, E. et al. The convolution of two differentiable functions on the circle need not be differentiable. Rev Mat Complut 32, 187–193 (2019). https://doi.org/10.1007/s13163-018-0274-5
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DOI: https://doi.org/10.1007/s13163-018-0274-5