Quaternion and fractional Fourier transform in higher dimension

摘要

Several quaternion Fourier transforms have received considerable attention in the last years. In this paper, based on the symplectic decomposition of quaternions, we prove that the Hermite functions in the spherical basis are the eigenfunctions of the QFTs. The relationship with Radon transform, the real and complex Paley-Wiener theorem, as well as the sampling formula for the 2-sided QFT are established in an easy way. Moreover, a two parameter fractional Fourier transform is introduced based on a new direct sum decomposition of the L2 space. The explicit formula and bound of the kernel are obtained. Compared with the Clifford-Fourier transform defined using eigenfunctions, the main adventure of this transform is that the integral kernel has a uniform bound for all dimensions.