Convolutions of Functions

Abstract

At the end of 2.3.9 we posed the problem of finding a binary operation on integrable functions that would correspond to pointwise multiplication of their Fourier transforms. To attempt directness by trying to define the result, say ƒ ✶ g, of applying this operation to functions ƒ, g ∈ L¹ by requiring that ${\left( {f * g} \right)^ \wedge } = \hat f \cdot \hat g$ is not very effective, because we do not know how to characterize A(Z) in such a way that it is clear that it is closed under pointwise multiplication. A more useful clue is provided by the orthogonality relations combined with the special properties of characters.