Frame Properties In Reproducing Kernel Spaces

Frame theory has been widely used in optics, signal processing, image processing, data compression, sampling theory and other fields up to the present. This thesis focuses on how to constructing frame sequence using reproducing kernel function in reproducing kernel spaces because reproducing kernel spaces have many excellent properties and reproducing kernel function plays an important role in these spaces. Since reproducing kernel function in these spaces have excellent properties, frame sequence with better forms can be constructed, which makes function decomposition and reconstruction in these spaces easy to implement. Main contents in this paper are as follows:The conditions of constructing Bessel sequence and Riesz-Fischer sequence using reproducing kernel function are given in reproducing kernel spaces. Hilbert space with complex value Bergman kernel is selected as an example, in the first place, Bessel sequence and Riesz-Fischer sequence are constructed by the reproducing kernel function. Both the Bessel sequence and the Riesz-Fischer sequence can be explicitly expressed by the Bergman kernel function; In the second place, some necessary conditions for a sequence to be Bessel sequence are given by virtue of the excellent properties of reproducing kernel spaces; Finally, frame sequence of Bergman space is given by Bessel sequence and Riesz-Fischer sequence.When the dual frame is given, any function in Bergman space can be reconstructed by the dual frame and the value of function at the frame sampling points using the properties of reproducing kernel.When some conditions are satisfied, the embedded subspaces of multi- resolution analysis and wavelet spaces can be proved to be reproducing kernel space, the reproducing kernel function can be constructed by wavelet function and scale function respectively, and the frame can be constructed by the reproducing kernel function. When the frame sampling points which satisfy some conditions are selected and one of the variables of the reproducing kernel function is selected as the frame sampling points, the reproducing kernel function forms the frame of these scale spaces. The invariance of perturbation is proved in the end.