Fourier transform and valuations (Q1630629)

For \(f \in L^1 (\mathbb{R}^n, \mathbb{C}),\) the Fourier transform of \(f\) is defined by \[ \mathcal{F}f(x)=\int_{\mathbb{R}^n} f(y) e^{-ix \cdot y} dy, \ x \in\mathbb{R}^n. \] A map \(z: L^1(\mathbb{R}^n, \mathbb{C}) \rightarrow C(\mathbb{R}^n, \mathbb{C})\) is \textit{positively \(\mathrm{GL}(n)\) covariant} if \(z(f\cdot \phi^{-1})(x)=|\det \phi |\ z(f) (\phi^t x)\) for every \(f \in L^1 (\mathbb{R}^n, \mathbb{C}),\ x \in \mathbb{R}^n\), and for all \(\phi \in \text{GL}(n)\) that have positive determinant. It is called \textit{logarithmic translation covariant} if \(z(f(\cdot -t))(x)=e^{-it\cdot x} z(f)(x),\) for every \(f \in L^{1}(\mathbb{R}^n, \mathbb{C}),\ x, t \in \mathbb{R}^n.\) It is called a \textit{valuation} if \(z(f \vee g)+z(f \wedge g)=z(f)+z(g)\) for all \(f, g \in L^{1}(\mathbb{R}^n, \mathbb{C}),\) where \(f \vee g\) denotes the pointwise maximum of \(f\) and \(g\) taken for the real and imaginary parts separately, and \(f \wedge g\) denotes the corresponding minimum of \(f\) and \(g\). In the paper under review the authors prove that the Fourier transform is a continuous, positively \(\mathrm{GL}(n)\) covariant and logarithmic translation covariant valuation. Moreover, the Fourier transform is only such a valuation on the complex valued functions.