The bounds of the odd dimensional Clifford-Fourier kernels

The even dimensional Clifford-Fourier transforms have been studied in detail in the last decade. However, the research on the odd dimension is hard to start because the closed expressions and the bounds of these kernels are not obtained. In this paper, we prove that the odd-dimensional Clifford-Fourier kernels are polynomially bounded as in the even-dimensional cases. The crucial ingredient in our proof is the closed expression of these kernels in the Laplace domain. With these bounds, various analytic properties can be established by carrying over the proof for even dimensions, such as inversion theorem and uncertainty principles.