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Two specific generalizations of the multidimensional Hilbert transform in Clifford analysis are constructed. It is shown that though in each of these generalizations some traditional properties of the Hilbert transform are inevitably lost, new bounded singular operators emerge on Hilbert or Sobolev spaces of L 2-functions

In this paper we devise a so-called cylindrical Fourier transform within the Clifford analysis context. The idea is the following: for a fixed vector in the image space the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector. For this Fourier kernel we now substitute a new Clifford-Fourier kernel such that, again… (More)

A new method for constructing Clifford algebra-valued orthogonal polynomials in the open unit ball of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clif-ford algebra-valued functions. The method consists in transforming the… (More)

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