On entire matrix-valued functions

Let A(z) be a given polynomial n×n matrix with det A(z)=1. It is shown that there exists an entire matrix-valued function X(z) such that expX(z)=A(z) if and only if the eigenvalues of A(z) are independent of z.     

Factorization of almost-periodic matrix-valued functions

Translated from Matematicheskie Zametki, Vol. 45, No. 6, pp. 74–82, June, 1989.     

Factorization of analytic matrix-valued functions

A new method is proposed for calculating the partial indices and the Wiener-Hopf factorization of analytic matrix-valued functions. It has computational advantages with respect to the method proposed by the author earlier. The new method is used to find the divisors of an analytic matrix-valued function a(t) that generate the zeros ofdet a (t). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 324–336, March, 1999.     

On periodic matrix-valued Weyl-Titchmarsh functions

We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl-Titchmarsh matrix-valued functions as well as a new version of the functional model for such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions, we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension which generate a periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach, we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.     

A density theorem for matrix-valued radial basis functions

Recently a new class of customized radial basis functions (RBFs) was introduced. We revisit this class of RBFs and derive a density result guaranteeing that any sufficiently smooth divergence-free function can be approximated arbitrarily closely by a linear combination of members of this class. This result has potential applications to numerically solving differential equations, such as fluid flows, whose solution is divergence free. AMS subject classification 41Axx, 41A30, 41A35, 41A60Svenja Lowitzsch: The results are part of the authorss dissertation written at Texas A&M University, College Station, TX 77843, USA.     

Estimate for the norm of the inverse of a differential operator

We obtain an estimate for the norm of the inverse of a differential operator on Lebesgue function spaces on the positive half-line.     

The algebra of harmonic functions for a matrix-valued transfer operator

We analyze matrix-valued transfer operators. We prove that the fixed points of transfer operators form a finite-dimensional C^∗-algebra. For matrix weights satisfying a low-pass condition we identify the minimal projections in this algebra as correlations of scaling functions, i.e., limits of cascade algorithms.     

Commutative and noncommutative representations of matrix-valued herglotz-nevanlinna functions

Operator realizations of matrix-valued Herglotz-Nevanlinna functions play an important and essential role in system theory, in the spectral theory of bounded nonselfadjoint operators, and in interpolation problems. Here, a generalization for realization results of the Brodskiǐ-Livsic type is given for Herglotz-Nevanlinna functions whose spectral measures are compactly supported.     

On the representation of a class of contractive matrix-valued functions

By a decomposition of L₊²(Cⁿ) in two orthogonal subspaces we obtain a representation of a matrix-valued function of the class $\text{J}$Π, defined by Arov (Darlington realization of matrix-valued functions, Izv. Akad. Nauk. SSSR, Ser. Mat. Tom.37 (1973), No. 6); (Math. USSR Izvestija7 (1973), No. 6, 1295–1326). Real matrix-valued functions of this class play an important role in methods of synthesis of scattering matrices of linear passive n-ports.     

Weighted norm inequalities for analytic functions

We present a weighted norm inequality involving convolutions of arbitrary analytic functions and certain confluent hypergeometric functions. This result implies a family of weighted norm inequalities both for entire functions of exponential type and for (generalized) hypergeometric series. The approach is based on author's general inequality for continuous functions and some hypergeometric transformations.     

Stability analysis of uncertain systems via matrix-valued Liapunov functions

In this paper, a new approach is given that is applied in the investigation of qualitative motion properties of systems whose parameters are prescribed unprecisely (uncertain systems). The general method of investigation is to construct sufficient conditions under which uncertain system motions possess a certain type of stability. In this work the main method of analysis for this class of systems is a generalized direct Lyapunov method based on matrix-valued auxiliary functions.     

Carathéodory theorem for matrix-valued functions and the maximal jump of spectral functions in extension problems

Translated from Matematicheskie Zametki, Vol. 48, No. 3, pp. 3–11, September, 1990.     

Variable Lebesgue norm estimates for BMO functions

In this paper, we are going to characterize the space BMO(? ⁿ ) through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space BMO(? ⁿ ) by using various function spaces. For example, Ho obtained a characterization of BMO(? ⁿ ) with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known John-Nirenberg inequality which can be seen as an extension to the variable Lebesgue spaces.     

A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer-Burmeister function

Based on a formula of Tseng, we show that the squared norm of the matrix-valued Fischer-Burmeister function has a Lipschitz continuous gradient. These authors' research was partially supported by Grants R314-000-057/042-112 of the National University of Singapore.