Research Of Frame Properties In Reproducing Kernel Hilbert Space

Frame theory is a new research direction with the rising of wavelet theory development, and it has promoted the development of functional analysis, operator theory and nonlinear approximate theory till present. Hilbert space as a class of inner product space is one of the important mathematical research objects. One of the purposes of the research of Hilbert space can be described abstractly as that any Hilbert space has standard orthogonal basis, and every element of Hilbert space can be said as linear combination of these family elements. So we can recognize the great importance of the basis of research space, but the required conditions is so strict that it is in want of agililty. Frame can easily resolve this problem. Frame is a series of elements which can satisfy the stability conditions. It can represent any single element in space in linearity, in the meanwhile, with strong agility. Frame elements may be linear correlated. In this sense, the frame can be regarded as generalized basis. Frame theory is widely applied in pure mathematics and applied mathematics.Reproducing kernel Hilbert space is a kind of important function space, and it is also the ideal space in the study of numerical analysis. It has fine numerical expression because a function existing in this space can get the discrete numerical issues expressed continuously so that optimization of all kinds of numerical problems become possible. So the properties of reproducing kernel Hilbert space are particularly important.This paper is aimed to carry out further research of the frame in reproducing kernel Hilbert space on the basis of existing researches and the whole project will basically divided in following three aspects: First, if there is a frame in reproducing kernel Hilbert space, its reproducing kernel function can be structured via a standard tight frame. Second, in reproducing kernel Hilbert space, a class of functions to be found may directly compute frame coefficient without the inverse of the frame operator. Third, if a frame with reproducing kernel form is known, a sequence is constructed via its Gram matrix, and the sufficient and necessary condition which become a frame is proved. A frame express is proved in Bergman space.