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A note on the convolution theorem for the Fourier transform

Charles S. Kahane (2011)

Czechoslovak Mathematical Journal

In this paper we characterize those bounded linear transformations T f carrying L 1 ( 1 ) into the space of bounded continuous functions on 1 , for which the convolution identity T ( f * g ) = T f · T g holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.

A pair of linear functional inequalities and a characterization of L p -norm

Dorota Krassowska, Janusz Matkowski (2005)

Annales Polonici Mathematici

It is shown that, under some general algebraic conditions on fixed real numbers a,b,α,β, every solution f:ℝ → ℝ of the system of functional inequalities f(x+a) ≤ f(x)+α, f(x+b) ≤ f(x)+β that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of L p -norm is given.

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