The Paley-Wiener theorem in \(\mathbb{R}^n\) with the Clifford analysis setting. (English) Zbl 1015.42013

Kou, Kit-Ian; Qian, Tao

J. Funct. Anal. 189, No. 1, 227-241 (2002).

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).

Reviewer: Terence Tao (Los Angeles)

MSC:

42B25

Maximal functions, Littlewood-Paley theory

Keywords:

Fourier analysis; Paley-Wiener theorem; conjugate harmonic system; Clifford analysis

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