On The Clifford-Fourier Transform

At present,there have been real Paley-wiener and Boas theorems of Quaterion and Clifford value function in the~2L space.On the basis of the study,in order to achieve further development,the two theorems can generalized to~pL space in the Quaderion and Clifford value function.First,under the classical fourier transform,based on the basic method of inversion theorem,we systematically study the real Paley-Wiener theorem of the Schwartz function and theL~p function.Then,we study the Fourier transform of Quaterion(the following is called QFT),and gives a series of related properties of QFT.Based on the Real Paley-Wiener theorem onL~2(R~2,H),inspired by[2]and[21],we think of extending the Real Paley-Wiener theorem fromL~2(R~2,H)toL~p(R~2,H).On the one hand,the convolution formula is not holding because of the noncommutative property of the Quaterion,we think of using constant approximation in harmonic analysis.on the other hand,at p?2,there is no classical plancherel theorem in QFT,we can adopt the idea of point-by-point approximation on the module.Finally,the Clifford is a domain wider than Quaternion.We first introduce the properties of the Fourier transform in Clifford(the following is called CFT),through the studying of the real Paley-Wiener and Boas theorems in[13],we think of extending the theorem toL~p function.Now we encounter two problems.one is the non-commutative of Clifford number,the convolution equation is not holding.On the other hand,at p?2,there was no corresponding plancherel theorem in CFT.In order to overcome the two difficulties,through Tuan's Real Paley-Wiener theorems of traditional Fourier transform,we can successfully avoid these problems by Hausdorff-Young andH(5)o(5)lderinequality,and some clever use of partial integration,so that we can extend the Paley-Wiener and Boas theorems from~2L space toL~p space in Clifford functions.