Research On Quaternion Fourier Transform

In this paper, we promote and prove the real Paley-Wiener theorems for the quater-nion Fourier transform on R~2 for Quaternion-valued Schwartz functions and LP-functions,and summarize the latest research results of the real Paley-Wiener theorem for scalar-and quaternion-valued L~2-functions.Firstly, in the elementary treatment based on the inversion theorem, we systemati-cally study the real Paley-Wiener theorem for Schwartz functions and the Lp-functions distributions for the Fourier transformRd.As an application, we show how to derive the real Paley-Wiener theorem via an approach that does not involve domain shifting. Then we first establish the real Paley-Wiener theorem for the Schwartz function.Secondly, we apply QFT to the quaternion field and discuss a series of related prop-erties of QFT in practical application. Different forms of QFT will give us different Plancherel theorem and Parseval theorem. These theorems play important role in the following theorem proving. Compared with the real Paley-Wiener theorem of classical Fourier transform, the real Paley-Wiener theorem of the quaternoin Fourier transform on L~2(R~2; H) can be more intuitively described as follows:the compactness of the support of Fourier transform by means of the norm of its partial derivative on R~2.Lastly, the classical Paley-Wiener theorem describes the Fourier transform of the function on L~2-space, the function of the theorem is an exponential type of entire function with support in the symmetry interval. This classical Paley-Wiener theorem has been used in a variety of transformations. Recently, we have a new idea of the Bang's real Paley-Wiener theorem. Inspired by Fu's papers, we want to extend the Paley-Wiener theorem from the L~2(R~2;H) to Lp(R~2;H). First we encounter a problem, due to lack of the Plancherel theorem for Lp-functions when p ? 2, we have to consider pointwise estimates as well as Lp-norms. And think of de Jeu's paper, the Paley-Wiener theorem on Lp(Rd;R), due to the fact that the quaternion algebra H is non-commutative. We can not extend the convolution properties of the classical Fourier transform to the quaternion domain. To overcome this difficulty, we introduce idea of approximation to the identity.Based on the study of the real Paley-Wiener theorem for the quaternion Fourier transform on L~2-space, we prove that the Paley-Wiener theorem is also established on the Lp-space.