On The Study Of Fractional Fourier Transform

As is known,the classical Fourier transform is a global transform,it can not express the local time-frequency properties of the signal.Therefore,a series of new transforms in signal analysis,such as wavelet transform and fractional Fourier transform,are proposed.In fact,fractional Fourier transform has been widely used in engineering.However,up to now,very few theoretical study in this topic was published.In this thesis,we will study the fractional Fourier transform from theoretical point of view.First of all,the fractional Fourier transforms are defined in many different ways.Inspired by some articles of S.Abdullah,and the relevant book"The Fractional Fourier Transform with Applications in Optics and Signal Processing"by H.M.Ozaktas et al,the definition of quaternion fractional Fourier transform(QFFT)is proposed.Secondly,we explore a series of related properties of QFFT,such as linearity,commutativity,combination,inversion formula,the relation with derivatives,and more.Finally,we consider the corresponding real Paley-Wiener theorem and Boas theorem for QFFT inL~p(R~2,H)space.The results are new in our literature.The first point is that we prove the partial derivative theorem for the QFFT.Furthermore,due to the fact that there is no convolution formula inL~p(R~2,H),so we use the approximation identity in harmonic analysis to solve this problem.And,at p?2,because there has not the Plancherel theorem,we use the pointwise estimate to solve it.In addition,the Boas theorem of QFFT is proved at the end of this thesis.