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.

Keywords:

• Fourier analysis
• Paley–Wiener theorem
• Exponential function
• Clifford algebra

Keywords:

• MSC 2000:
• 30G35

Notes

†Dedicated to Richard Delanghe on the occasion of his 65th birthday.