Plane waves, biregular functions and hypercomplex Fourier analysis. (English) Zbl 0597.30059

Let \(\Omega\) be an open subset of \(R^{m+1}\times R^{m+1}\). Then a function \(f\in C_ 1(\Omega)\) with values in the complex Clifford algebra \(C_ m\) is called biregular in \(\Omega\) if f satisfies the system \(D_ xf(x,t)=f(x,t)D_ t=0,\) where \(D_ x\) and \(D_ t\) are Dirac type operators in \(R^{m+1}\) [see also F. Brackx and W. Pincket, ibid. 9, 21-35 (1985; reviewed above)].
In this paper we give a formula for the biregular extension (i.e. the Cauchy-Kowalewski extension) of a given real analytic function f(x,t) in \(R^ m\times R^ m\) and apply it to the Fourier kernel \(\exp (i<x,t>)\). More in general, we study the biregular extensions of plane waves of the form \(f(<x,t>)\), leading to a system of eight differential equations in five dimensions.
Finally, we use the complexified biregular extension \({\mathcal E}(\tau,z)\) of \(\exp (i<t,x>)\) to define a Fourier-Borel type transform for the analytic functionals of Clifford analysis [see also F. Brackx, R. Delanghe and F. Sommen, Cliifford analysis (1982; Zbl 0529.30001)] and to prove Paley-Wiener type theorems for this transform.


30G35 Functions of hypercomplex variables and generalized variables
33B10 Exponential and trigonometric functions
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46F15 Hyperfunctions, analytic functionals