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A characterization of Fourier transforms

Philippe Jaming (2010)

Colloquium Mathematicae

The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ℤ/nℤ, the integers ℤ, the torus 𝕋 and the real line. We also ask a related question for the twisted convolution.

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from L p ( ) to L q ( ) . We also give a condition on p which is necessary if this operator maps L p ( ) into L²().

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