Octonionic Hypercomplex Analysis And Its Application

In this thesis,we investigate the octonionic hypercomplex analysis and its application.The octonion plays an important role in the G2 geometry and particle physics.This has also inspired the recent rapid development of octonionic hypercomplex analysis.Aiming at several key problems of octonionic hypercomplex analysis,in this paper we make an in-depth and detailed investigation.This includes the following aspects:1.The relationship between hypercomplex analysis and complex analysis.2.Octonionic Fourier transform and the real Paley-Wiener theorem for octonionic Fourier transform.3.Octonionic Witt basis in the theory of Octonionic Hermitian analysis.4.Octonionic Hilbert space.Our main results are as follows:1.We have built a new bridge between quaternionic hypercomplex analysis and complex analysis.We show that the Cauchy integral formula for a monogenic function for which its range contains in the complex plane of quaternion space,turns out to be the Bochner-Martinelli integral formula for an associated holomorphic functions on C2.2.For the octonion Fourier transform,we establish the real Paley-Wiener theorem.It relates the mean of derivatives of a function with the support of its octonion Fourier transform via (?) for any octonion-valued Sobolev function f ∈ H∞(R3,O).3.The Witt basis in the complex plane plays a key role in the construction of the complex variables and complex operators in place of the real variables and real operators.In this thesis,we introduce the Witt basis on the tensor product of several octonionic variables with Clifford algebras.This construction depends heavily on a finite subgroup of the octonionic automorphism involving binary expression in Z8.The construction of octonionic Witt basis should be the corner stone to the octonionic Hermitian analysis.4.The octonionic left module theory should be the starting point of the octonionic Hilbert spaces,we build the isomorphism between Cl7 left modules category Cl7-Mod and left octonionic modules category(?)-Mod.With the help of the isomorphism,we provide a complete classification of left octonionic modules.