Basis And Frame In Reproducing Kernel Space

Frame is a tool that elements of Hilbert space, it can be seen as the promotion of the basement, and basement is different because the frame does not require the element of linearly independent; it is decomposition coefficients determined. In terms of the nature, the frame elements of the space are more natural and more extensive than the elements of the space.As frame of the Riesz basis with a certain redundancy, it can make the wavelet coefficients in the low accuracy. They have rebuild at a relatively high accuracy, so in recent years on the frame of wavelet research studies have become one of the hottest researches. At present, the frame is one of the important tools, not only in the widely used and dynamic mathematical research, but also in the image processing, digital communications, information science, especially in the signal processing in the signal de-noising, feature extraction, processing the application of the increasingly widespread. Main contents in this paper are as follows:First, it can be proved that the wavelet space and the multi-resolution analysis's subspace, using regeneration property of the reproducing kernel space, are reproducing kernel space under the controlled conditions. It was constructed by reproducing kernel function and scaling function, and also reproducing kernel space frame structure, we also constructed a frame expression of reproducing kernel form under the presented frame based on wavelet reproducing kernel Hilbert space. When it specifically obtains the higher and lower bound of the frame, then it provides a great convenience for the signal decomposition and reconstruction frame.Second, when it starts from the operator of the frame, for the reproducing kernel space, the relationship between the standard orthogonal basis and the frame can be discussed the substrate. When using the reproducing kernel function, it can be discussed the property of the frame, and the construction algorithm of the reproducing kernel space frame.It is beneficial for the space's numerical analysis and function approximation, which is constructed in the frame of this article. It can provide not only a good function, but also the reference value in the construction of the frame space.