An elementary proof of the Paley–Wiener theorem in C m †

Abstract

We give an elementary proof of the Paley–Wiener theorem in several complex variables using Clifford algebra. Thanks to the extension of the exponential function , where x is extended to , and ξ is extended to C m , the proof of the theorem in one complex variable based on analytic continuation is closely followed to give a proof for several complex variables. This shows that with the help of Clifford algebras multi-variable cases may be treated similarly as the single variable case. †Dedicated to Richard Delanghe on the occasion of his 65th birthday.