Cyclic AFD algorithm for the best rational approximation

We propose a practical algorithm of best rational approximation of a given order to a function in the Hardy H² space on the unit circle and on the real line. The type of approximation is proved to be equivalent with Blaschke form approximation. The algorithm is called Cyclic adaptive Fourier decomposition as it adaptively selects one parameter for each cycle on the basis of the maximal selection principle proved in the literature of adaptive Fourier decomposition. Copyright © 2013 John Wiley & Sons, Ltd.     

Two‐dimensional adaptive Fourier decomposition

One‐dimensional adaptive Fourier decomposition, abbreviated as 1‐D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szegö and higher‐order Szegö kernels of the context. In the present paper, we study multi‐dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product‐TM Systems; and the other uses Product‐Szegö Dictionaries. With the Product‐TM Systems approach, we prove that at each selection of a pair of parameters, the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product‐Szegö dictionary approach, we show that pure greedy algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre‐orthogonal Greedy Algorithm (P‐OGA). We prove its convergence and convergence rate estimation, allowing a weak‐type version of P‐OGA as well. The convergence rate estimation of the proposed P‐OGA evidences its advantage over orthogonal greedy algorithm (OGA). In the last part, we analyze P‐OGA in depth and introduce the concept P‐OGA‐Induced Complete Dictionary, abbreviated as Complete Dictionary. We show that with the Complete Dictionary P‐OGA is applicable to the Hardy H² space on 2‐torus. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive Fourier decomposition‐based Dirac type time‐frequency distribution

The Dirac‐type time‐frequency distribution (TFD), regarded as ideal TFD, has long been desired. It, until the present time, cannot be implemented, due to the fact that there has been no appropriate representation of signals leading to such TFD. Instead, people have been developing other types of TFD, including the Wigner and the windowed Fourier transform types. This paper promotes a practical passage leading to a Dirac‐type TFD. Based on the proposed function decomposition method, viz., adaptive Fourier decomposition, we establish a rigorous and practical Dirac‐type TFD theory. We do follow the route of analytic signal representation of signals founded and developed by Garbo, Ville, Cohen, Boashash, Picinbono, and others. The difference, however, is that our treatment is theoretically throughout and rigorous. To well illustrate the new theory and the related TFD, we include several examples and experiments of which some are in comparison with the most commonly used TFDs. Copyright © 2016 John Wiley & Sons, Ltd.     

Strong convergence of two‐dimensional Vilenkin‐Fourier series

We prove that certain means of the quadratical partial sums of the two‐dimensional Vilenkin‐Fourier series are uniformly bounded operators from the Hardy space to the space for We also prove that the sequence in the denominator cannot be improved.     

Intrinsic mono‐component decomposition of functions: An advance of Fourier theory

We propose a function decomposition model, called intrinsic mono‐component decomposition (IMD). It is a continuation of the recent study on adaptive decomposition of functions into mono‐components (MCs). It is a further improvement of two recent results of which one is adaptive decomposition of functions into modified inner functions, and the other is decomposition by using adaptive Takenaka‐Malmquist systems. The proposed new decomposition model is of less restriction and thus gains more adaptivity. The theory is valid to both the unit circle and the real line contexts. Copyright © 2009 John Wiley & Sons, Ltd.     

A novel 2D partial unwinding adaptive Fourier decomposition method with application to frequency domain system identification

This paper proposes a two‐dimensional (2D) partial unwinding adaptive Fourier decomposition method to identify 2D system functions. Starting from Coifman in 2000, one‐dimensional (1D) unwinding adaptive Fourier decomposition and later a type called unwinding AFD have been being studied. They are based on the Nevanlinna factorization and a maximal selection. This method provides fast‐converging rational approximations to 1D system functions. However, in the 2D case, there is no genuine unwinding decomposition. This paper proposes a 2D partial unwinding adaptive Fourier decomposition algorithm that is based on algebraic transforms reducing a 2D case to the 1D case. The proposed algorithm enables rational approximations of real coefficients to 2D system functions of real coefficients. Its fast convergence offers efficient system identification. Numerical experiments are provided, and the advantages of the proposed method are demonstrated.     

Reproducing kernel approximation in weighted Bergman spaces: Algorithm and applications

In this paper, we present algorithms of preorthogonal adaptive Fourier decomposition (POAFD) in weighted Bergman spaces. POAFD, as has been studied, gives rise to sparse approximations as linear combinations of the corresponding reproducing kernels. It is found that POAFD is unavailable in some weighted Hardy spaces that do not enjoy the boundary vanishing condition; as a result, the maximal selections of the parameters are not guaranteed. We overcome this difficulty with two strategies. One is to utilize the shift operator while the other is to perform weak POAFD. In the cases when the reproducing kernels are rational functions, POAFD provides rational approximations. This approximation method may be used to 1D signal processing. It is, in particular, effective to some Hardy H^p space functions for p not being equal to 2. Weighted Bergman spaces approximation may be used in system identification of discrete time‐varying systems. The promising effectiveness of the POAFD method in weighted Bergman spaces is confirmed by a set of experiments. A sequence of functions as models of the weighted Hardy spaces, being a wider class than the weighted Bergman spaces, are given, of which some are used to illustrate the algorithm and to evaluate its effectiveness over other Fourier type methods.     

Solving a fourth‐order fractional diffusion‐wave equation in a bounded domain by decomposition method

In this article, the Adomian decomposition method has been used to obtain solutions of fourth‐order fractional diffusion‐wave equation defined in a bounded space domain. The fractional derivative is described in the Caputo sense. Convergence of the method has been discussed with some illustrative examples. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On the Cauchy problem for the two‐component b‐family system

In this paper, we establish the local well‐posedness for the two‐component b‐family system in a range of the Besov space. We also derive the blow‐up scenario for strong solutions of the system. In addition, we determine the wave‐breaking mechanism to the two‐component Dullin–Gottwald–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

A matrix rational Lanczos method for model reduction in large‐scale first‐ and second‐order dynamical systems

In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model‐order reduction problems using multipoint moment matching‐based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos‐like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second‐order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given.     

Symmetry of two‐parameter homogeneous transition function

In this paper, we study a very interesting problem, that is, symmetry of the two parameters s,t>0 for a two‐parameter homogeneous transition function and obtain some important and interesting results.     

Total probability of two‐parameter generalized transition function

In this paper, we research some properties of the total probability of a two‐parameter generalized transition function and obtain some important and interesting results.     

Constructing rational and multi‐wave solutions to higher order NEEs via the Exp‐function method

In this paper, we present an application of some known generalizations of the Exp‐function method to the fifth‐order Burgers and to the seventh‐order Korteweg de Vries equations for the first time. The two examples show that the Exp‐function method can be an effective alternative tool for explicitly constructing rational and multi‐wave solutions with arbitrary parameters to higher order nonlinear evolution equations. Being straightforward and concise, as pointed out previously, this procedure does not require the bilinear representation of the equation considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive Fourier decomposition of functions in quaternionic Hardy spaces

We study decompositions of functions in the Hardy spaces into linear combinations of the basic functions in the orthogonal rational systems B_n, which are obtained in the respective contexts through Gram–Schmidt orthogonalization process on shifted Cauchy kernels. Those lead to adaptive decompositions of quaternionic‐valued signals of finite energy. This study is a generalization of the main results of the first author's recent research in relation to adaptive Takenaka–Malmquist systems in one complex variable. Copyright © 2011 John Wiley & Sons, Ltd.     

On a second‐order expansion of the truncated singular subspace decomposition

We present a second‐order expansion for singular subspace decomposition in the context of real matrices. Furthermore, we show that, when some particular assumptions are considered, the obtained results reduce to existing ones. Some numerical examples are provided to confirm the theoretical developments of this study. Copyright © 2016 John Wiley & Sons, Ltd.     

On the successive supersymmetric rank‐1 decomposition of higher‐order supersymmetric tensors

In this paper, a successive supersymmetric rank‐1 decomposition of a real higher‐order supersymmetric tensor is considered. To obtain such a decomposition, we design a greedy method based on iteratively computing the best supersymmetric rank‐1 approximation of the residual tensors. We further show that a supersymmetric canonical decomposition could be obtained when the method is applied to an orthogonally diagonalizable supersymmetric tensor, and in particular, when the order is 2, this method generates the eigenvalue decomposition for symmetric matrices. Details of the algorithm designed and the numerical results are reported in this paper. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

In this study, with the aid of Wolfram Mathematica 11, the modified exp ‐expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well‐known nonlinear evolutionary equation, namely; the two‐component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two‐component second order KdV evolutionary system with the finite forward difference method by using the Fourier‐Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of and norm error. We present the comparison between the exact and numerical solutions of the two‐component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp ‐expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models.