A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed. The scheme is third-order accurate in time and O(2−jp) accurate in space. Unlike Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving non-linear problems. The compactly supported orthogonal wavelet bases D6 developed by Daubechie...

An orthogonal Haar scattering transform is a deep network computed with a hierarchy of additions, subtractions and absolute values over pairs of coefficients. Unsupervised learning optimizes Haar pairs to obtain sparse representations of training data with an algorithm of polynomial complexity. For signals defined on a graph, a Haar scattering is computed by cascading orthogonal Haar wavelet tr...

An orthogonal basis of L which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F . Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and recovering than they do in any other orthogonal basis. In fact, simple thresholding in an uncondit...

We generalize the windowed Fourier transform to the variable-windowed Fourier transform. This generalization brings the Gabor transform and the wavelet transform under the same framework. Using frame theory we characterize frames and orthonormal bases for the variable windowed Fourier series (VWFS). These characterizations are formulated explicitly in terms of window functions. Therefore they c...

Let Sp(n,R) be the symplectic group, and K∗ n its maximal compact subgroup. Then G = Sp(n,R)/K∗ n can be realized as the Siegel domain of type one. The square-integrable representation of G gives the admissible wavelets AW and wavelet transform. The characterization of admissibility condition in terms of the Fourier transform is given. The Bergman kernel follows from the viewpoint of coherent s...

In this work, a class of multivariate multiwavelet packages with multi-scale are introduced, which are generalizations of multivariant wavelet packages. A method for designing a sort of biorthogonal multiwavelet packages in higher dimensions is presented and their biorthogonality property is characterized by virtue of time-frequency analysis method, matrix theory, and operator theory. Three bio...

This paper addresses a Compensatory Wavelet Neuro-Fuzzy System (CWNFS) for temperature control. The proposed CWNFS model is five-layer structure, which combines the traditional Takagi-Sugeno-Kang (TSK) fuzzy model and the wavelet neural networks (WNN). We adopt the non-orthogonal and compactly supported functions as wavelet neural network bases. Besides, the compensatory fuzzy reasoning method ...

Several authors have demonstrated that significant improvements can be obtained in wavelet-based signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Bo...

We consider implementation of operators via filter banks in the framework of the multiresolution analysis. Our method is particularly efficient for convolution operators. Although our method of applying operators to functions may be used with any wavelet basis with a sufficient number of vanishing moments, we distinguish two particular settings, namely, orthogonal bases and the autocorrelation ...