BIORTHOGONAL UNCONDITIONAL BASES OF COMPACTLY SUPPORTED MATRIX VALUED WAVELETS

In this paper, we introduce the concept of biorthogonal matrix valued wavelets. We elaborate on perfect reconstruction matrix filter banks which are assembled by matrix FIR fllters and we deduce that the resulting matrix valued wavelet functions have compact support. Moreover, we form biorthogonal unconditional bases for the space of matrix valued signals. To validate the theory, a class of biorthogonal and orthonormal matrix valued wavelets are given. The connection of the present scheme with the theory of multiwavelets are also explored.     

Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

Stability of best rational Chebyshev approximation

In this paper we consider best Chebyshev approximation to continuous functions by generalized rational functions using an optimization theoretical approach introduced in [[5.]]. This general approach includes, in a unified way, usual, weighted, one-sided, unsymmetric, and also more general rational Chebychev approximation problems with side-conditions. We derive various continuity conditions for the optimal value, for the feasible set, and the optimal set of the corresponding optimization problem. From these results we derive conditions for the upper semicontinuity of the metric projection, which include some of the results of Werner [On the rational Tschebyscheff operator, Math. Z. 86 (1964), 317–326] and Cheney and Loeb [On the continuity of rational approximation operators, Arch. Rational Mech. Anal. 21 (1966), 391–401].     

Operators Which Do Not Have the Single Valued Extension Property

In this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator T  L(X) on a Banach space X and some quantities associated with T which play an important role in Fredholm theory. In particular, we shall consider some conditions for which T does not have the single valued extension property at a point λ_o  C.     

A parametric smooth variational principle and support properties of convex sets and functions

We show a modified version of Georgiev's parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D(∂h) is pathwise connected and R(∂h) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R(∂h) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets.     

Iteratively reweighted algorithm for signals recovery with coherent tight frame

We consider the problem of compressed sensing with a coherent tight frame and design an iteratively reweighted least squares algorithm to solve it. To analyze the problem, we propose a sufficient null space property under a tight frame (sufficient D‐NSP). We show that, if a measurement matrix A satisfies the sufficient D‐NSP of order s, then an s‐sparse signal under the tight frame can be exactly recovered. Furthermore, if A satisfies the restricted isometric property with tight frame D of order 2bs, then it also satisfies the sufficient D‐NSP of order as with a < b and b sufficiently large. We prove the convergence of the algorithm based on the sufficient D‐NSP and give the upper error bounds. In numerical experiments, we use the discrete cosine transform, discrete Fourier transform, and Haar wavelets to verify the effectiveness of this algorithm. With increasing measurement number, the signal‐to‐noise ratio increases monotonically.     

A simultaneous decomposition of a matrix triplet with applications

Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A=±A*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A?BXB*?CYC* and then solve two conjectures on the maximal and minimal possible ranks of A?BXB*?CYC* with respect to X=±X* and Y=±Y*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB* + CYC*=A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Inequalities for predictive ratios and posterior variances in natural exponential families

The predictive ratio is considered as a measure of spread for the predictive distribution. It is shown that, in the exponential families, ordering according to the predictive ratio is equivalent to ordering according to the posterior covariance matrix of the parameters. This result generalizes an inequality due to Chaloner and Duncan who consider the predictive ratio for a beta-binomial distribution and compare it with a predictive ratio for the binomial distribution with a degenerate prior. The predictive ratio at x₁ and x₂ is defined to be p_g(x₁)p_g(x₂)/[p_g( )]² = h_g(x₁, x₂), where p_g(x₁) = ∫ ƒ(x₁θ) g(θ) dθ is the predictive distribution of x₁ with respect to the prior g. We prove that h_g(x₁, x₂) ≥ h_g^*(x₁, x₂) for all x₁ and x₂ if ƒ(xθ) is in the natural exponential family and Cov_gx(θ) ≥ Cov_g^*x(θ) in the Loewner sense, for all x on a straight line from x₁ to x₂. We then restrict the class of prior distributions to the conjugate class and ask whether the posterior covariance inequality obtains if g and g^* differ in that the “sample size”     

Neumann and mixed problems on curvilinear polyhedra

We prove regularity results inL ^p Sobolev spaces. On one hand, we state some abstract results byL ^p functional techniques: exponentially decreasing estimates in dyadic partitions of cones and dihedra, operator valued symbols and Marcinkievicz's theorem. On the other hand, we derive more concrete statements with the help of estimates about the first non-zero eigenvalue of some Laplace-Beltrami operators on spherical domains.     

Real valued iterative methods for solving complex symmetric linear systems

Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi‐definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time‐stepping integration schemes based on Padé approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.     

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)     

On verifiable sufficient conditions for sparse signal recovery via <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> minimization

We discuss necessary and sufficient conditions for a sensing matrix to be “s-good”—to allow for exact ℓ ₁-recovery of sparse signals with s nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect ℓ ₁-recovery (nonzero measurement noise, nearly s-sparse signal, near-optimal solution of the optimization problem yielding the ℓ ₁-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse ℓ ₁-recovery and to efficiently computable upper bounds on those s for which a given sensing matrix is s-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties.     

Differential approximation results for the traveling salesman problem with distances 1 and 2

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min_TSP12 and max_TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard-approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any ϵ>0, it is NP-hard to approximate both problems better than within 741/742+ϵ. The same results hold when dealing with a generalization of min_ and max_TSP12, where instead of 1 and 2, edges are valued by a and b.     

Edge detection of noisy images based on cellular neural networks

This paper studies a technique employing both cellular neural networks (CNNs) and linear matrix inequality (LMI) for edge detection of noisy images. Our main work focuses on training templates of noise reduction and edge detection CNNs. Based on the Lyapunov stability theorem, we derive a criterion for global asymptotical stability of a unique equilibrium of the noise reduction CNN. Then we design an approach to train edge detection templates, and this approach can detect the edge precisely and efficiently, i.e., by only one iteration. Finally, we illustrate performance of the proposed methodology from the aspect of peak signal to noise ratio (PSNR) through computer simulations. Moreover, some comparisons are also given to prove that our method outperforms classical operators in gray image edge detection.     

Common Principal Components for Dependent Random Vectors

Let the kp-variate random vector X be partitioned into k subvectors X_i of dimension p each, and let the covariance matrix Ψ of X be partitioned analogously into submatrices Ψ_ij. The common principal component (CPC) model for dependent random vectors assumes the existence of an orthogonal p by p matrix β such that β^tΨ_ijβ is diagonal for all (i, j). After a formal definition of the model, normal theory maximum likelihood estimators are obtained. The asymptotic theory for the estimated orthogonal matrix is derived by a new technique of choosing proper subsets of functionally independent parameters.     

Deconvolution of Point Sources: A Sampling Theorem and Robustness Guarantees

In this work we analyze a convex-programming method for estimating superpositions of point sources or spikes from nonuniform samples of their convolution with a known kernel. We consider a one-dimensional model where the kernel is either a Gaussian function or a Ricker wavelet, inspired by applications in geophysics and imaging. Our analysis establishes that minimizing a continuous counterpart of the ℓ₁-norm achieves exact recovery of the original spikes as long as (1) the signal support satisfies a minimum-separation condition and (2) there are at least two samples close to every spike. In addition, we derive theoretical guarantees on the robustness of the approach to both dense and sparse additive noise. © 2018 Wiley Periodicals, Inc.     

A posteriori error analysis of the heterogeneous multiscale method for homogenization problems

In this Note we derive a posteriori error estimates for a multiscale method, the so-called heterogeneous multiscale method, applied to elliptic homogenization problems. The multiscale method is based on a macro-to-micro formulation. The macroscopic method discretizes the physical problem in a macroscopic finite element space, while the microscopic method recovers the unknown macroscopic data on the fly during the macroscopic stiffness matrix assembly process. We propose a framework for the analysis allowing to take advantage of standard techniques for a posteriori error estimates at the macroscopic level and to derive residual-based indicators in the macroscopic domain for adaptive mesh refinement. To cite this article: A. Abdulle, A. Nonnenmacher, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Eigenvalues and automorphisms of a graph

Let G be the automorphism group of a graph Γ and let λ be an eigenvalue of the adjacency matrix of Γ. In this article, (i) we derive an upper bound for rank(G), (ii) if G is vertex transitive, we derive an upper bound for the extension degree of ?(λ) over ?, (iii) we study automorphism groups of graphs without multiple eigenvalues, (iv) we study spectra of quotient graphs associated with orbit partitions.     

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦⁿW)⁽ⁿ⁾W^-1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.     

Stochastic boundary value problems: a white noise functional approach

Summary We give a program for solving stochastic boundary value problems involving functionals of (multiparameter) white noise. As an example we solve the stochastic Schrödinger equation {ie391-1} whereV is a positive, noisy potential. We represent the potentialV by a white noise functional and interpret the product of the two distribution valued processesV andu as a Wick productV u. Such an interpretation is in accordance with the usual interpretation of a white noise product in ordinary stochastic differential equations. The solutionu will not be a generalized white noise functional but can be represented as anL ¹ functional process.