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.     

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.     

Comparison of adaptive mono-component decompositions

As presented in some recent publications, adaptively choosing the parameters of a Takenaka–Malmquist (TM) system according to the given signal gives rise to the so called adaptive Fourier decomposition (AFD). Besides optimal selections of the parameters to ensure the maximal energy gain at each step, AFD produces entries of non-negative analytic instantaneous frequency. The latter enables us to define a reasonable time–frequency distribution with most desired properties. The energy principle together with the unwinding process through factorizing out the inner function factors yields two variations of the standard AFD, both having appeared in the literature. They are referred by this paper as, respectively, unwinding adaptive Fourier decomposition (UAFD) and double sequence unwinding adaptive Fourier decomposition (DSUAFD). After a short summary of the three adaptive decompositions, the present paper makes comparison between them, as well as with the traditional Fourier series decomposition (FD). The related Dirac type time–frequency distributions associated with mono-components and mono-component decompositions of multi-components are introduced with examples. As necessary preparation we recall the concept mono-component and related knowledge in the introduction section.     

Fast algorithm of adaptive Fourier series

Adaptive Fourier decomposition (AFD, precisely 1‐D AFD or Core‐AFD) was originated for the goal of positive frequency representations of signals. It achieved the goal and at the same time offered fast decompositions of signals. There then arose several types of AFDs. The AFD merged with the greedy algorithm idea, and in particular, motivated the so‐called pre‐orthogonal greedy algorithm (pre‐OGA) that was proven to be the most efficient greedy algorithm. The cost of the advantages of the AFD‐type decompositions is, however, the high computational complexity due to the involvement of maximal selections of the dictionary parameters. The present paper constructs one novel method to perform the 1‐D AFD algorithm. We make use of the FFT algorithm to reduce the algorithm complexity, from the original to , where N denotes the number of the discretization points on the unit circle and M denotes the number of points in [0,1). This greatly enhances the applicability of AFD. Experiments are performed to show the high efficiency of the proposed algorithm.     

Construction of rigid local systems and integral representations of their sections

We give a method for constructing all rigid local systems of semi‐simple type, which is different from the Katz–Dettweiler–Reiter algorithm. Our method follows from the construction of Fuchsian systems of differential equations with monodromy representations corresponding to such local systems, which give an explicit solution of the Riemann–Hilbert problem. Moreover, we show that every section of such local systems has an integral representation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An alternating projections method for certain linear problems in a Hilbert space

Recent extensions of von Neumann's alternating projection methodpermit the computation of proximity projections onto certainconvex sets. This paper exploits this fact in constructing aglobally convergent method for minimizing linear functions overa convex set in a Hilbert space. In particular, we solve theeducational testing problem and an inverse eigenvalue problem,two difficult problems involving positive semidefiniteness constraints.     

Investigations on the approximability and computability of the Hilbert transform with applications

It was recently shown that on a large class of important Banach spaces there exist no linear methods which are able to approximate the Hilbert transform from samples of the given function. This implies that there is no linear algorithm for calculating the Hilbert transform which can be implemented on a digital computer and which converges for all functions from the corresponding Banach spaces. The present paper develops a much more general framework which also includes non-linear approximation methods. All algorithms within this framework have only to satisfy an axiom which guarantees the computability of the algorithm based on given samples of the function. The paper investigates whether there exists an algorithm within this general framework which converges to the Hilbert transform for all functions in these Banach spaces. It is shown that non-linear methods give actually no improvement over linear methods. Moreover, the paper discusses some consequences regarding the Turing computability of the Hilbert transform and the existence of computational bases in Banach spaces.     

Computing genus 2 curves from invariants on the Hilbert moduli space

We give a new method for generating genus 2 curves over a finite field with a given number of points on the Jacobian of the curve. We define two new invariants for genus 2 curves as values of modular functions on the Hilbert moduli space and show how to compute them. We relate them to the usual three Igusa invariants on the Siegel moduli space and give an algorithm to construct curves using these new invariants. Our approach simplifies the complex analytic method for computing genus 2 curves for cryptography and reduces the amount of computation required.     

用广义交互确认方法选择良好参数进行多元多项式散乱数据自然样条光顺

1．简介多元样条函数在多元逼近中发挥很大作用，已有数量相当多的综合报告和研究论文正式发表，就在1996年6月在法国召开的第三届国际曲线与曲面会议上便有不少多元样条方面的报告，不过总的感觉是仍然缺乏对噪声数据特别是散乱数据的有效光顺方法．李岳生、崔锦泰、关履泰、胡日章等讨论广义调配样条与张量积函数，并用希氏空间样条方法处理多元散乱数据样条插值与光顺，提出多元多项式自然样条，推广了相应一元的结果．我们知道，在样条光顺中有一个如何选择参数的问题，用广义交互确认方法（generalizedcross－validation，以下简称GC…     

Copulae,Self-Affine Functions,and Fractal Dimensions

We use the fact that the functions defined on the unit interval whose graphs support a copula are those that are Lebesgue-measure-preserving in order to characterize self-affine functions whose graphs are the support of a copula. This result allows computation of the Hausdorff, packing, and box-counting dimensions. The discussion is applied to classic examples such as the Peano and Hilbert curves, and the results are extended to discontinuous self-affine functions.     

Closed range positive operators on Banach spaces

A bounded positive operator on a Hilbert space has closed range if and only if the operator and its square root have common ranges. We give an extension of this result for positive operators acting on reflexive Banach spaces. Some other results concerning positive operators on Hilbert spaces are carried over to this general case.     

N solutions for a derivative nonlinear Schrödinger‐type equation via Riemann‐Hilbert approach

The inverse scattering transform for the derivative nonlinear Schrödinger‐type equation is studied via the Riemann‐Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann‐Hilbert problem is established for the derivative nonlinear Schrödinger‐type equation. In the inverse scattering process, N‐soliton solutions of the derivative nonlinear Schrödinger‐type equation are obtained by solving Riemann‐Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.     

Self‐adjointness of perturbed biharmonic operators on Riemannian manifolds

We give a sufficient condition for the essential self‐adjointness of a perturbed biharmonic‐type operator acting on sections of a Hermitian vector bundle on a geodesically complete Riemannian manifold with Ricci curvature bounded from below by a (possibly unbounded) non‐positive function depending on the distance from a reference point. We also establish the separation property in the case when the corresponding operator acts on functions.     

On mathematical modelling of absorption and dispersion of seismic waves for low frequencies

The problem of determining the dispersion and absorption of plane seismic waves by a given seismic quality factor is dealt with in the case that the quality factor tends to a positive limit as the frequency goes to zero. General formulae are derived for the phase velocity and the attenuation coefficient by solving the corresponding Riemann–Hilbert problems for analytic functions in the upper half-plane.     

Local spectral theory for normal operators in Krein spaces

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.     

Extrapolation methods for fixed‐point multilinear PageRank computations

Nonnegative tensors arise very naturally in many applications that involve large and complex data flows. Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher order power method is one of the most commonly used technique for the computation of positive Z‐eigenvectors of this type of tensors. However, unlike the matrix case, the method may fail to converge even for irreducible tensors. Moreover, when it converges, its convergence rate can be very slow. These two drawbacks often make the computation of the eigenvectors demanding or unfeasible for large problems. In this work, we consider a particular class of nonnegative tensors associated with the multilinear PageRank modification of higher order Markov chains. Based on the simplified topological ε‐algorithm in its restarted form, we introduce an extrapolation‐based acceleration of power method type algorithms, namely, the shifted fixed‐point method and the inner‐outer method. The accelerated methods show remarkably better performance, with faster convergence rates and reduced overall computational time. Extensive numerical experiments on synthetic and real‐world datasets demonstrate the advantages of the introduced extrapolation techniques.     

关于算子正矩阵

关于算子正矩阵侯晋川，高明杵（山西师范大学数学系，临汾０４１００４）基金项目：国家自然科学基金、山西省自然科学基金资助课题１９９２年３月１９日收到．由于应用的广泛性，人们对于寻求块方阵（）为半正定矩阵的（充分或必要）条件一直很感兴趣，且已获得许多深刻...     

Aveiro method in reproducing kernel Hilbert spaces under complete dictionary

Aveiro method is a sparse representation method in reproducing kernel Hilbert spaces, which gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying reproducing kernel Hilbert space. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro method. To avoid those difficulties, we propose an new Aveiro method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so‐called pre‐orthogonal greedy algorithm involving completion of a given dictionary. The new method is called Aveiro method under complete dictionary. The complete dictionary consists of all directional derivatives of the underlying reproducing kernels. We show that, under the boundary vanishing condition bring available for the classical Hardy and Paley‐Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro method and the greedy algorithm.     

Spectrum of the Kerzman–Stein operator for a family of smooth regions in the plane

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on the boundary. For bounded regions with smooth boundary, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions. Here we give an explicit computation of its Hilbert–Schmidt norm for a family of simply connected regions. We also give an explicit computation of the Cauchy operator acting on an orthonormal basis, and we give estimates for the norms of the Kerzman–Stein and Cauchy operators on these regions. The regions are the first regions that display no apparent Möbius symmetry for which there now is explicit spectral information.     

Schwarz methods for inequalities with contraction operators

We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.