The Paley-Wiener theorem in \(\mathbb{R}^n\) with the Clifford analysis setting (Q1597998)

The classical Paley-Wiener theorem asserts that an \(L^2\) function on \(\mathbb R\) has Fourier transform restricted to an interval \([-R,R]\) if and only if its holomorphic extension to the complex plane exists and grows by at most \(O(\exp(R|z|)\)). The authors extend this theorem to higher dimensions, proving that an \(L^2\) function on \(\mathbb R^n\) has Fourier transform restricted to the ball of radius \(R\) if and only if its (Clifford-valued) left-monogenic extension to \(\mathbb R \times \mathbb R^n\) exists and grows by at most \(O(\exp(R|z|))\). Furthermore, one can extend the Fourier inversion formula to \(\mathbb R \times \mathbb R^n\) by means of the left-monogenic extensions of the complex exponential. As a corollary, one obtains a very similar theorem for conjugate harmonic systems (since they are the components of left-monogenic functions).