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convolution; Fourier transform
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.
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