KARHUNEN-LOèVE Expansions And Reproducing Kernel Hilbert Spaces Of Several Types Of Gaussian Processes

In probability and statistics, the Gaussian process is a common and important stochastic process, and reproducing kernel Hilbert space is an important theory of computational mathematics. Karhunen-Loève expansion and reproducing kernel Hilbert space of Gaussian process have closely connected with many fields, such as compact operator, positive definite function, Green fuction, conformal mapping, orthogonal basis, support vector machine, small ball estimate, matrix, spectral decomposition, kernel function. The research of Karhunen-Loève expansion and reproducing kernel Hilbert space of Gaussian process allows us to grasp exactly the series expansions of some Gaussian processes and the coefficients of the series is not relevant, we can also use the covariance function of Guaussian process to construct reproducing kernel Hilbert space. This will enable us to solve more flexibly and conveniently the problems of the subject areas, such as the wavelet transform, random process handling, signal processing, machine learning.The topic of"Karhunen-Loève expansions and the reproducing kernel Hilbert spaces of several types of Gaussian processes"is cross-cutting areas of research topic of Gaussian random process and the reproducing kernel Hilbert space, and mainly for zero mean Gaussian process, by using Mercer's theorem and the reproducing kernel Hilbert space theory we yield Karhunen-Loève expansions and the reproducing kernel Hilbert spaces of three types of Gaussian processes. This paper mainly studies the following problems of two aspects:(1) Karhunen-Loève expansion of Gaussian process: For zero mean Gaussian process, firstly we use Mercer's theorem to yield Karhunen-Loève expansions of the first and the second type of Gaussian process; Then we yield Karhunen-Loève expansion of the third type of Gaussian process by use of Mercer's theorem and the properties of Bessel function. (2) The reproducing kernel Hilbert space of Gaussian process: Firstly we expand and complete the section of the reproducing kernel to construct the reproducing kernel Hilbert space; Secondly we use Mercer's theorem and have characteristic function as the base function to construct the reproducing kernel Hilbert space; Finally we yield the corresponding reproducing kernel Hilbert spaces of three types of Gaussian processes.The structure of this paper is as follows: In the introduction of the first chapter we introduce the relevant circumstances of this topic; In the second chapter we study the Karhunen-Loève expansion of the first type of Gaussian process; The Karhunen-Loève expansion of the second type of Gaussian process is yielded in the third chapter; In the fourth chapter we yield the Karhunen-Loève expansion of the third type of Gaussian process; We construct the reproducing kernel Hilbert space of Gaussian process in the last chapter.