Some Theories And Applications Of The Reproducing Kernel Hilbert Space

Reproducing kernel theory has important application in numerical analysis, differen-tial equation, probability and statistics and so on. Recently, these great theoretical devel-opments about the theory of Reproducing kernel Hilbert space(RKHS) also have impor-tant practical implications, but such methodologies face serious shortcomings because theRKHS framework is not sufficient.This thesis firstly investigates the constructions of the RKHS with some restrainedfunctions, then proposes a method to compute the reproducing kernel on the reproducingkernelspacewithweightedintegral, andshowssomeprinciplesaboutthespacewithmultiinterzones. At last the thesis studies applications of the RKHS in the fields of differentialequations, inverse problems and image processing. Main contributions of this dissertationare as follows:(1) Based on the function theory, the construction of the RKHS with some restrainedfunctions is presented by constructing a orthogonal projection from the common space tothe new one. Then the constructions also be studied by the difference theory of the RKHS.(2) This dissertation provides a new definition for reproducing kernel space withweighted integral and presents a general framework to construct and calculate the repro-ducing kernel for the space. Furthermore, some principles about the space with multiinterzones are presented.(3) Based on the framework of the RKHS with some restrained functions, many dif-ferential equations boundary value problems have been studied based on the reproducingkernel space, such as, second-order initial and boundary value problems,(k+j)-pointboundary value problems and four-point nonlinear fractional integro-differential equa-tions.(4)This dissertation studies the theory of the multivariate RKHS based on the crossproduct theory of RKHS. At the same time, the dissertation presents the method com-bined the RKHS method with the multiscale method for solving the inverse heat sourceproblems.(5)Based on the reproducing property, a novel approach combined the RKHS withimage interpolation method is presented．The proposed image-interpolating method is performed in the multivariate RKHS．...