Estimation of matrix valued realized signal to noise ratio

In this paper we derive several estimators of matrix valued realized signal to noise ratio as defined by Khatri and Rao (1987, IEEE Trans. Acoust. Speech Signal Process. ASSP-35, No. 5 671–679) for real and complex cases. To do so we define the matrix valued confluent hypergeometric distribution and establish some of its properties. Also we derive unique admissible estimates under generalized Pitman nearness. Finally a discussion of confidence interval estimation for signal to noise ratio is given.     

Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems

We present a new family of compactly supported and symmetric biorthogonal wavelet systems. Each refinement mask in this family has tension parameter ω. When ω = 0, it becomes the minimal length biorthogonal Coifman wavelet system (Wei et al., IEEE Trans Image Proc 7:1000–1013, 1998). Choosing ω away from zero, we can get better smoothness of the refinable functions at the expense of slightly larger support. Though the construction of the new biorthogonal wavelet systems, in fact, starts from a new class of quasi-interpolatory subdivision schemes, we find that the refinement masks accidently coincide with the ones by Cohen et al. (Comm Pure Appl Math 45:485–560, 1992, §6.C) (or Daubechies 1992, §8.3.5), which are designed for the purpose of generating biorthogonal wavelets close to orthonormal cases. However, the corresponding mathematical analysis is yet to be provided. In this study, we highlight the connection between the quasi-interpolatory subdivision schemes and the masks by Cohen, Daubechies and Feauveau, and then we study the fundamental properties of the new biorthogonal wavelet systems such as regularity, stability, linear independence and accuracy.     

On realization of the Kreĭn–Langer class Nκ of matrix‐valued functions in Pontryagin spaces

In this paper the realization problems for the Kre?n–Langer class N_κ of matrix‐valued functions are being considered. We found the criterion when a given matrix‐valued function from the class N_κ can be realized as linear‐fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodskii–Livsic rigged operator colligation) with the main operator acting on a rigged Pontryagin space Π_κ with indefinite metric. We specify three subclasses of the class N_κ (R) of all realizable matrix‐valued functions that correspond to different properties of a realizing system, in particular, when the domains of the main operator of a system and its conjugate coincide, when the domain of the hermitian part of a main operator is dense in Π_κ . Alternatively we show that the class N_κ (R) can be realized as transfer matrix‐functions of some canonical impedance systems with self‐adjoint main operators in rigged spaces Π_κ . The case of scalar functions of the class N_κ (R) is considered in details and some examples are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

M-Band Burt–Adelson Biorthogonal Wavelets

For every integer M>2 we introduce a new family of biorthogonal MRAs with dilation factor M, generated by symmetric scaling functions with small support. This construction generalizes Burt–Adelson biorthogonal 2-band wavelets. For M{3,4} we are able to find simple explicit expressions for two different families of wavelets associated with these MRAs: one with better localization and the other with interesting symmetry–antisymmetry properties. We study the regularity of our scaling functions by determining their Sobolev exponent, for every value of the parameter and every M. We also study the critical exponent when M=3.     

Recurrence relation for biorthogonal polynomials

We use a geometric approach to obtain a recurrence relation for two families of biorthogonal polynomials associated to a nonsingular, strongly regular matrix M. We propose a “look-ahead procedure” for computing the biorthogonal polynomials when M has singular or ill-conditioned leading principal submatrices. These polynomials lead to two recursive triangular factorizations for the inverse of a nonsingular matrix M which is not necessarily strongly regular.     

A necessary and sufficient condition for the biorthogonality of {ρ 1,ρ 2}

In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functionsρ ₁ andρ ₂. The functions are useful in the theory of biorthogonal wavelet.     

The properties of a class of biorthogonal vector-valued nonseparable bivariate wavelet packets

In this paper, we introduce a class of vector-valued wavelet packets of space L²(R²,C^κ), which are generalizations of multivariate wavelet packets. A procedure for constructing a class of biorthogonal vector-valued wavelet packets in higher dimensions is presented and their biorthogonality properties are characterized by virtue of matrix theory, time–frequency analysis method, and operator theory. Three biorthogonality formulas regarding these wavelet packets are derived. Moreover, it is shown how to gain new Riesz bases of space L²(R²,C^κ) from these wavelet packets. Relation to some physical theories such as the Higgs field is also discussed.     

具有高消失矩的3-进双正交对称小波的构造

本文用矩阵对称扩充来构造了具有高消失矩的3带对称双正交小波.利用矩阵扩充,获得了3维矩阵对称扩充方法和小波构造的算法,并且,该算法便于计算机程序化实现;利用两个实例验证了相关的结论.     

Construction and decomposition of biorthogonal vector-valued wavelets with compact support

In this article, we introduce vector-valued multiresolution analysis and the biorthogonal vector-valued wavelets with four-scale. The existence of a class of biorthogonal vector-valued wavelets with compact support associated with a pair of biorthogonal vector-valued scaling functions with compact support is discussed. A method for designing a class of biorthogonal compactly supported vector-valued wavelets with four-scale is proposed by virtue of multiresolution analysis and matrix theory. The biorthogonality properties concerning vector-valued wavelet packets are characterized with the aid of time–frequency analysis method and operator theory. Three biorthogonality formulas regarding them are presented.     

Comparison theory for Riccati equations

We give a short new proof for the comparison theory of the matrix valued Riccati equationB′+B ²+R=0 with singular initial values. Applications to Riemannian geometry are briefly indicated.     

Vector valued Riesz distributions on Euclidian Jordan algebras

Let V be a Euclidean Jordan algebra, Гthe associated symmetric cone and G be the identity component of the linear automorphism group of Г.In this paper we associate to a certain class of spherical representations (ρ, ɛ) of G certain ɛ-valued Riesz distributions generalizing the classical scalar valued Riesz distributions on V. Our construction is motivated by the analytic theory of unitary highest weight representations where it permits to study certain holomorphic families of operator valued Riesz distributions whose positive definiteness corresponds to the unitarity of a representation of the automorphism group of the associated tube domain Г +iV.     

N–Term Approximation in Anisotropic Function Spaces

In L₂(0, 1)²) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.     

Q-functions of Hermitian Contractions of Krein-Ovcharenko Type

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space ℌ. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Q_μ- and Q_M-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Q_μ- and Q_M-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.     

B-Browder spectra and localized SVEP

In this paper we study the relationships between the B-Browder spectra and some other spectra originating from Fredholm theory and B-Fredholm theory. This study is done by using the localized single valued extension property. In particular, we shall see that many spectra coincide in the case that a bounded operator T, or its dual T*, or both, admits the single valued extension property.      

Reproducing Kernels and Conformal Mappings in

In this paper we look at the theory of reproducing kernels for spaces of functions in a Clifford algebra _{0, n}. A first result is that reproducing kernels of this kind are solutions to a minimum problem, which is a non-trivial extension of the analogous property for real and complex valued functions. In the next sections we restrict our attention to Szegö and Bergman modules of monogenic functions. The transformation property of the Szegö kernel under conformal transformations is proved, and the Szegö and Bergman kernels for the half space are calculated.     

Zeros of Fredholm Operator Valued Hp–Functions

This paper is concerned with Fredholm operator valued H^p – functions on the unit disc, where the Fredholm operators action a Banach space. Sufficient conditions are presented which guarantee that Fatou's theorem is valid. Using the theory of traces and determinants on quasi – Banach operator ideals, we develop conditions that guarantee that the zeros of Fredholm operator valued H^p – functions satisfy the Blaschke condition.     

Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol

We perform a spectral analysis of the preconditioned Hermitian/skew‐Hermitian splitting (PHSS) method applied to multilevel block Toeplitz linear systems in which the coefficient matrix T_n(f) is associated with a Lebesgue integrable matrix‐valued function f. When the preconditioner is chosen as a Hermitian positive definite multilevel block Toeplitz matrix T_n(g), the resulting sequence of PHSS iteration matrices M_n belongs to the generalized locally Toeplitz class. In this case, we are able to compute the symbol ?(f,g) describing the asymptotic eigenvalue distribution of M_nwhen n→∞ and the matrix size diverges. By minimizing the infinity norm of the spectral radius of the symbol ?(f,g), we are also able to identify effective PHSS preconditioners T_n(g) for the matrix T_n(f). A number of numerical experiments are presented and commented, showing that the theoretical results are confirmed and that the spectral analysis leads to efficient PHSS methods. Copyright © 2015 John Wiley & Sons, Ltd.     

The average number of critical rank-one approximations to a tensor

Motivated by the many potential applications of low-rank multi-way tensor approximations, we set out to count the rank-one tensors that are critical points of the distance function to a general tensor v. As this count depends on v, we average over v drawn from a Gaussian distribution, and find a formula that relates this average to problems in random matrix theory.     

Degrees of nonlinearity in forbidden 0–1 matrix problems

A 0–1 matrix A is said to avoid a forbidden 0–1 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of Davenport–Schinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turán-type subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and non-geometric algorithms.Clearly a 0–1 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Füredi and Hajnal conjectured that if P corresponds to an acyclic graph then the maximum weight (number of 1s) in an n×n matrix avoiding P is O(nlogn). In the first part of the article we refute of this conjecture. We exhibit n×n matrices with weight Θ(nlognloglogn) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 0–1 matrices. In the second part of the article we simplify one aspect of Keszegh and Geneson’s proof that there are infinitely many minimal nonlinear forbidden 0–1 matrices. In the last part of the article we investigate the relationship between 0–1 matrices and generalized Davenport–Schinzel sequences. We prove that all forbidden subsequences formed by concatenating two permutations have a linear extremal function.