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Keywords:
convolution; Fourier transform
convolution; Fourier transform
Summary:
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.
References:
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[2] Krantz, S.: A Panorama of Harmonic Analysis, The Carus Mathematical Monographs. Number 27, The Mathematical Association of America, Washington D.C. (1999). MR 1710388
[3] Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York, 1st edition (1966). MR 0210528 | Zbl 0142.01701
[4] Stein, E. M., Shakarchi, R.: Real Analysis. Princeton University Press, Princeton, New Jersey (2005). MR 2129625 | Zbl 1081.28001
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