In this paper, we study the invertibility of sequences consisting of finitely many bounded linear operators from a Hilbert space to others. We show that a sequence of operators is left invertible if and only if it is a g-frame. Therefore, our result connects the invertibility of operator sequences with frame theory.
We build Wiener measure for the path space on the Heisenberg group by using of the heat kernel corresponding to the sub-Laplacian and give the definition of the Wiener integral.Then we give the FeynmanKac formula.
We study the approximation of the inverse wavelet transform using Riemannian sums.We show that when the Fourier transforms of wavelet functions satisfy some moderate decay condition,the Riemannian sums converge to the function to be reconstructed as the sampling density tends to infinity.We also study the convergence of the operators introduced by the Riemannian sums.Our result improves some known ones.
The problem of two order statistics detection schemes for the detection of a spatially distributed target in white Gaussian noise are studied.When the number of strong scattering cells is known,we first show an optimal detector,which requires many processing channels.The structure of such optimal detector is complex.Therefore,a simpler quasi-optimal detector is then introduced.The quasi-optimal detector,called the strong scattering cells’ number dependent order statistics（SND-OS） detector,takes the form of an average of maximum strong scattering cells with a known number.If the number of strong scattering cells is unknown in real situation,the multi-channel order statistics（MC-OS） detector is used.In each channel,a various number of maximums scattered from target are averaged.Then,the false alarm probability analysis and thresholds sets for each channel are given,following the detection results presented by means of Monte Carlo simulation strategy based on simulated target model and three measured targets.In particular,the theoretical analysis and simulation results highlight that the MC-OS detector can efficiently detect range-spread targets in white Gaussian noise.
The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.
Let be the quaternion Heisenberg group, and let P be the affine automorphism group of . We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2（ ）. A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on . . A Semyanistyi-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth. In addition, we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on .
