On The Study Of Special Relativistic Space--time Fourier Transform

At present,the theory of the classical Fourier transform is relatively complete.In recent decades,the Fourier transform in hypercomplex analysis has received considerable attention.This type of Fourier transform processes a multipath signal into a quaternion-or Clifford algebra-valued signal containing the multipath signal.This thesis considers a special relative space-time Fourier transform(hereinafter referred to as SFT)in the Clif-ford algebra Cl_(3,1),which is a transformation proposed and discussed by E.Hitzer from a mathematical point of view in recent years.First,this thesis modifies the SFT proposed by E.Hitzer and studies its basic prop-erties,such as Plancherel's theorem,inverse formula,partial derivative theorem,etc.Second,inspired by real Paley-Wiener theorem on the space of quaternion-valued p-square integrable functions,we think of extending the real Paley-Wiener theorem from L^p(R~2,H)to L^p(R^(3,1),Cl_(3,1)).Due to the non-commutative property of space-time al-gebra,many classical methods can not be used.For example,the convolution formula does not hold any more.We overcome these difficulties through the joint application of multiple inequalities,and give the real Paley-Wiener theorem and Boas theorem of SFT in L^p-case.Finally,through the study of Roe's theorem of the classical case and quaternion-Fourier transform,this thesis generalizes Roe's theorem to space-time algebra.Due to the non-commutative properties of space-time algebra,we use H?lder's inequality,partial dif-ferential theorem and patchwork elimination to solve the difficulties in our proof.Finally,we establish the Roe's theorem of SFT in two ways.