Closed Expression Of The Hypercomplex Fourier Kernel

Fourier transform is the core concept of the branch of mathematics called Fourier analysis.Classical Fourier transform not only has deep connections with partial differ-ential equation,number theory,representation theory,mathematical physics,but also has found innumerable applications in both engineering,and has become a fundamental tool for scientific research.Clifford analysis makes it possible to construct a genuine multi-dimensional Fourier transform in contrast to the already mentioned traditional approach which consists of taking tensor products of one-dimensional phenomena.In recent years,quite a bit of attention has been paid to the Clifford-Fourier trans-form.This transform is further generalized using the representation theory of the Lie algebra sl2 and group symmetry.A complete classification of the integral transforms with similar properties is given.However,only the kernels of few special cases are obtained.The integral transforms related with Lie algebra sl2 include Dunkl transform,the(k,a)-generalized Fourier transform and its generalization in the Clifford setting as well.The explicit expressions of these integral kernels still need to be determined.Especially,the expression of the Dunkl kernel has attracted quite a lot interests in the past 30 years,because it has deep connection with Markov process,random matrices,Calogero-Moser-Sutherland system,the minimal representation of conformal groups.In this thesis,we develop a new method based on the Laplace transform to study the closed expression of the hyper-complex Fourier kernel.This method is further developed to compute the Dunkl dihedral kernel and the(k,a)-Fourier kernel,as well as the bounds of these kernels.The main results of this thesis are as follows.1.The kernel of the Clifford-Fourier transform and its generalization in the Laplace domain is obtained.We equally obtain the plane wave decomposition and find new inte-gral representations for the kernel in all dimensions.For the even dimension case,poly-nomial bounds for the kernel functions are obtained.We define and compute the formal generating function for the even dimensional kernels.2.As the different geometry properties between Euclidean space and the hyperbolic space,we introduce a new generalization of the Helgason-Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models.The explicit integral kernels of even dimensions are derived.Furthermore we establish the formal generating function of the even dimensional kernels.In the computations,fractional integration plays a key unifying role.3.For the Dunkl kernel and the(κ,a)-Fourier kernel,by making use of the Pois-son kernel,we compute the closed expression of the generalized Fourier kernel in the Laplace domain.The integral expression of the Dunkl kernel in the dihedral setting and the(0,a)-generalized Fourier kernel is obtained by inverse Laplace transform.In case the parameters involved are integers,explicit formulas are obtained.New bounds for the ker-nel of the(0,2/n)-generalized Fourier transform are obtained as well using mathematical induction and the technique of special function.