This thesis represents an attempt to show the theory and applications of Quaternion analysis. Quaternions are widespread in the real world, such as the representation of rotations in three-dimensional space, and three channels color images. This dissertation focuses on two topics in terms of Quaternions. They are the energy concentration problems and color face recognition problem under the framework of Quaternions. The first part is about the prolate spheroidal wave functions. One of the fundamental problems in communications is finding the energy distribution of signals in time and frequency domains. The most energy concentration real-valued signal is the prolate spheroidal wave function. In Chapter 3, we review the one dimensional prolate spheroidal wave functions and apply them to signal recovery and feature extraction problems. In Chapter 4 we discuss the energy concentration problem in the framework of Quaternions and find the energy distribution of quaternion-valued signals in time and frequency domains. In this chapter, we give a novel kind of hypercomplex signals (quaternion-valued signals) whose energy concentration is maximal in both time and frequency under quaternion Fourier transform and quaternion linear canonical transforms domains. These signals are called the quaternionic prolate spheroidal wave functions. We present their definitions and properties and show that they can reach the extreme case in energy concentration problem both from the theoretical and experimental description. In particular, quaternionic prolate spheroidal wave functions are shown as an effective tool for signal extrapolation problem. The color face recognition problems under the framework of Quaternions are introduced in this Chapter 5. We study the collaborative representation-based classification (CRC) and sparse representation-based classification (SRC) methods in face recognition (FR). They are originally designed in the real setting for grayscale image based FR. They represent the color channels of a query color image separately and ignore the structural correlation information among the color channels. To remedy this limitation, in this work we propose two novel representation-based classification methods for color face recognition, namely quaternion collaborative representation-based classification (QCRC) and quaternion sparse representation-based classification (QSRC). ii By modeling each color image as a pure quaternion-valued signal, they naturally preserve the color structures of both query and gallery color images while uniformly coding the query channel images in a holistic manner. Despite the empirical success of CRC and SRC on FR, few theoretical results are developed to guarantee their effectiveness. Another purpose of this work is to establish the theoretical guarantee for QCRC and QSRC under mild conditions. Comparisons with competing methods on benchmark real-world databases consistently show the superiority of the proposed methods for color face recognition problem.