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The generalized sampling expansions associated with quaternion Fourier transform

English Abstract

The Whittaker-Shannon-Kotel’nikov (WSK) sampling theorem provides a reconstruction formula for the Paley-Wiener class of bandlimited functions. In this thesis, we would like to develop the sampling expansions of quaternion-valued functions in two different directions. On the one hand, we investigate some important properties of quaternion Fourier transform (QFT) such as inversion, Plancherel theorem, convolution theorem. We present the Papoulis-like generalized sampling expansion associated with QFT by using generalized translation and convolution. It is shown that a σ-bandlimited function in QFT sense can be reconstructed in terms of the samples of output functions of M linear systems based on QFT. Some examples are presented to illustrate the result. On the other hand, a generalization of the WSK sampling theorem is extended by using the theory of quaternion reproducing kernel Hilbert space. Also, we present a novel approach, based on the quaternion reproducing kernel Hilbert space, to the generalization of the prolate spheroidal wave functions. This generalization is then employed to obtain a sampling formula for a general class of bandlimited functions. A special case of our result is to show that the 2D generalized prolate spheroidal wave functions obtained by Slepian can also be used to achieve a sampling series for functions bandlimited to the cube. The solutions to the energy concentration problems in quaternion Hilbert space with QFT are also investigated.

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Cheng, Dong


Faculty of Science and Technology




Fourier analysis


Mathematics -- Department of Mathematics


Kou, Kit Ian

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