Plancherel theorem and quaternion Fourier transform for square integrable functions

The quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on the properties of QFT. The classical convolution theorem and multiplication formula are only suitable for 2D Fourier transform of complex-valued signal, and do not hold for QFT of quaternion-valued signal. The purpose of this paper is to overcome these problems and establish the Plancherel and inversion theorems of QFT in the square integrable signals space L2. First, we investigate the behaviours of QFT in the integrable signals space L1. Next, we deduce the energy preservation property which extends functions from L1 to L2 space. Moreover, some other important properties such as modified multiplication formula are also analyzed for QFT.Comment: 20 pages, 0 figure

Sampling expansions associated with quaternion difference equations

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of discrete transform, we form a quaternion function space whose elements have sampling expansions. Moreover, through formulating boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and polynomials with quaternion coefficients. We show that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are eigenvalues of the matrix. Several examples are given to illustrate the results

Trapped Resonant Fermions above Superfluid Transition Temperature

We investigate trapped resonant fermions with unequal populations within the local density approximation above the superfluid transition temperature. By tuning the attractive interaction between fermions via Feshbach resonance, the system evolves from weakly interacting fermi gas to strongly interacting fermi gas, and finally becomes bose-fermi mixture. The density profiles of fermions are examined and compared with experiments. We also point out the simple relationships between the local density, the axial density, and the gas pressure within the local density approximation.Comment: 9 pages, 4 figure

Development of serial verb constructions in Cantonese- speaking preschool children

"A dissertation submitted in partial fulfilment of the requirements for the Bachelor of Science (Speech and Hearing Sciences), The University of Hong Kong, June 30, 2006."Also available in print.Thesis (B.Sc)--University of Hong Kong, 2006.published_or_final_versionSpeech and Hearing SciencesBachelorBachelor of Science in Speech and Hearing Science