Abstract
Denote by \({\bf R}^n_1\) the real-linear span of \({\bf e}_0, {\bf e}_1, \ldots , {\bf e}_n\), where \({\bf e}_0 = 1, {\bf e}_i {\bf e}_j + {\bf e}_j {\bf e}_i =-2\delta_{ij}, 1 \leq i, j \leq n.\) Under the concept of left-monogeneity defined through the generalized Cauchy-Riemann operator we obtain the direct sum decomposition of \(L^2({\bf R}^n_1), n > 1,\)
where \(\Omega k\) is the right-Clifford module of finite linear combinations of functions of the form \(R(x)h(\vert x\vert)\), where, for \(d = n + 1\), the function R is a k- or \(-(d + \vert k\vert - 2)\)-homogeneous leftmonogenic function, for \(k > 0\) or \(k < 0\), respectively, and h is a function defined in [0,∞) satisfying a certain integrability condition in relation to k, the spaces \(\Omega k\) are invariant under Fourier transformation. This extends the classical result for \(n = 1\). We also deduce explicit Fourier transform formulas for functions of the form \(R(x)h(r)\) refining Bochner’s formula for spherical k-harmonics.
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Fei, Mg., Qian, T. Direct Sum Decomposition of L2(Rn 1) into Subspaces Invariant under Fourier Transformation. J Fourier Anal Appl 12, 145–155 (2006). https://doi.org/10.1007/s00041-004-4058-6
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DOI: https://doi.org/10.1007/s00041-004-4058-6