Abstract
In this paper we characterize those bounded linear transformations Tf carrying L 1(ℝ1) into the space of bounded continuous functions on ℝ1, for which the convolution identity T(f * g) = Tf · Tg holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Járai: Measurable solutions of functional equations satisfied almost everywhere. Math. Pannon. 10 (1999), 103–110.
S. Krantz: A Panorama of Harmonic Analysis, The Carus Mathematical Monographs. Number 27, The Mathematical Association of America, Washington D.C., 1999.
W. Rudin: Real and Complex Analysis. McGraw-Hill, New York, 1st edition, 1966.
E.M. Stein and R. Shakarchi: Real Analysis. Princeton University Press, Princeton, New Jersey, 2005.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kahane, C.S. A note on the convolution theorem for the Fourier transform. Czech Math J 61, 199–207 (2011). https://doi.org/10.1007/s10587-011-0006-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-011-0006-1