Recently, kernel methods for nonlinear processing have been gained widely attention. The reproducing kernel Hilbert space is the fundamental in this method. No adaptive kernel has been developed, so far, for complex valued signals. Moreover, the complex reproducing kernels are used in an increasing number of machine learning problems. According this approach, we develop Quaternion Reproducing Kernel (QRKS) and Quaternion Reproducing Kernel Hilbert Space (QRKHS). We provide a general framework to attack the problem of adaptive filtering of quaternion signals, using both methods. In addition, we modified HR calculus with the concept of inner product in quaternion Hilbert space. The quaternion gradient is provided in quaternion Hilbert spaces. For solving optimization problems, one common approach is to gain the gradient of the objective function. Simple rules, such as product rule and chain rule is obtained in the novel manner in the further study. Finally the quaternion kernel least-mean-square (QKLMS) algorithm is also presented.
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