Reproducing Kernel Spaces Based On Windowed Fourier Transform

The windowed Fourier transform is an effective tool of signal analysis in image processing, and it has been widely used in practical applications. Since its image spaces do not be paid enough attention and its properties do not be studied deeply. This thesis intensively focuses on windowed Fourier transform image spaces and gets that the image spaces are reproducing kernel Hilbert spaces proceeding from this problem. At the same time we further study the properties of windowed Fourier transform image spaces, The main results in this paper are as follows:Windowed Fourier transform from L² (R) to L² (R²) is studied in this paper. The results that windowed Fourier transform is tightly combined with reproducing kernel and windowed Fourier transform image spaces which are reproducing kernel spaces are proved.For different windowed function, the Gauss wavelet is chosen as the windowed function, then windowed Fourier transform is done and the image spaces is got that is a reproducing kernel Hilbert spaces. A new reproducing kernel function with analytic expression and corresponding kernel spaces T₀ L² (R ) are constructed making use of reproducing kernel special technique; Further, the Hermite functions are chosen as windowed function, then the windowed Fourier transform is done, meanwhile, the more common reproducing kernel function with analytic expression and the corresponding reproducing kernel Hilbert spaces T_n L² (R )are constructed, then the relation of two reproducing kernel functions in reproducing kernel Hilbert spaces T_n L² (R )and T₀ L² (R )is given.Operators of reproducing kernel spaces are studied as an application of reproducing kernel theory in the thesis, then the intrinsic properties of abstract operators are shown by means of studying concrete operators. Thereby the localization operators in reproducing kernel spaces T_n L² (R ) are considered and the eigenvalues and eigenfunctions are worked out, at the same time a Hilbert function spaces is made orthogonal decomposition using reproducing kernel spaces T_n L² (R )which is shown.