Geodesic
A note on the convolution theorem for the Fourier transform
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 199-207.

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In this paper we characterize those bounded linear transformations $Tf$ carrying $L^{1}( \mathbb {R}^{1}) $ into the space of bounded continuous functions on $\mathbb {R}^{1}$, for which the convolution identity $T(f\ast g) =Tf\cdot Tg $ holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.
DOI : 10.1007/s10587-011-0006-1
Classification : 39B22, 42A38, 47B33, 47B38
Keywords: convolution; Fourier transform 
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Kahane, Charles S. A note on the convolution theorem for the Fourier transform. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 199-207. doi : 10.1007/s10587-011-0006-1. http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0006-1/

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