scholarly journals An elementary proof of the polynomial matrix spectral factorization theorem

2014 ◽  
Vol 144 (4) ◽  
pp. 747-751 ◽  
Author(s):  
Lasha Ephremidze
Keyword(s):  
Unit Circle ◽  
Linear Algebra ◽  
Simple Proof ◽  
Real Line ◽  
Complex Analysis ◽  
Elementary Proof ◽  
Polynomial Matrix ◽  
Spectral Factorization ◽  
Factorization Theorem ◽  
Matrix Spectral Factorization

A very simple proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line), which relies on elementary complex analysis and linear algebra, is presented.

Rank-deficient spectral factorization and wavelets completion problem

2015 ◽  
Vol 13 (03) ◽  
pp. 1550013 ◽  
Author(s):  
L. Ephremidze ◽  
I. Spitkovsky ◽  
E. Lagvilava
Keyword(s):  
Polynomial Matrix ◽  
Elementary Solution ◽  
Spectral Factorization ◽  
Factorization Theorem ◽  
Constructive Proof ◽  
Completion Problem ◽  
Matrix Spectral Factorization

A simple constructive proof of polynomial matrix spectral factorization theorem is presented in the rank-deficient case. It is then used to provide an elementary solution to the wavelets completion problem.

scholarly journals An Analytic Proof of the Matrix Spectral Factorization Theorem

2008 ◽  
Vol 15 (2) ◽  
pp. 241-249
Author(s):  
Lasha Ephremidze ◽  
Gigla Janashia ◽  
Edem Lagvilava
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Spectral Factorization ◽  
Factorization Theorem ◽  
Matrix Functions ◽  
Analytic Proof ◽  
Wiener’S Theorem ◽  
The Matrix ◽  
Matrix Spectral Factorization

Abstract An analytic proof of Wiener's theorem on factorization of positive definite matrix-functions is proposed.

Matrix Completions, Moments, and Sums of Hermitian Squares

2011 ◽  
Author(s):  
Mihály Bakonyi ◽  
Hugo J. Woerdeman
Keyword(s):  
Linear Algebra ◽  
Undergraduate Students ◽  
Complex Analysis ◽  
Measure Theory ◽  
Complex Function ◽  
Complex Function Theory ◽  
Extensive Discussion ◽  
The Subject ◽  
Processing Control ◽  
Developing Area

Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly developing area of mathematics and applications and gives complete proofs of many recently solved problems. With MATLAB codes and more than two hundred exercises, the book is ideal for a special topics course for graduate or advanced undergraduate students in mathematics or engineering, and will also be a valuable resource for researchers. Often driven by questions from signal processing, control theory, and quantum information, the subject of this book has inspired mathematicians from many subdisciplines, including linear algebra, operator theory, measure theory, and complex function theory. In turn, the applications are being pursued by researchers in areas such as electrical engineering, computer science, and physics. The book is self-contained, has many examples, and for the most part requires only a basic background in undergraduate mathematics, primarily linear algebra and some complex analysis. The book also includes an extensive discussion of the literature, with close to six hundred references from books and journals from a wide variety of disciplines.

Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

A Short Proof of an Interpolation Theorem

1974 ◽  
Vol 17 (1) ◽  
pp. 127-128 ◽  
Author(s):  
Edward Hughes
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Simple Proof ◽  
Real Line ◽  
Short Proof ◽  
Interpolation Theorem ◽  
Operator Interpolation ◽  
Upper Half Plane ◽  
Positive Functions ◽  
Class Of Functions

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

ASYMPTOTIC EXPANSIONS FOR RIEMANN–HILBERT PROBLEMS

2008 ◽  
Vol 06 (03) ◽  
pp. 269-298 ◽  
Author(s):  
W.-Y. QIU ◽  
R. WONG
Keyword(s):  
Asymptotic Expansion ◽  
Complex Analysis ◽  
Elementary Proof ◽  
Hilbert Problem ◽  
Jump Condition ◽  
Similar Type ◽  
Matrix Solution ◽  
Jump Matrix ◽  
The Matrix ◽  
Riemann Hilbert Problem

Let Γ be a piecewise smooth contour in ℂ, which could be unbounded and may have points of self-intersection. Let V(z, N) be a 2 × 2 matrix-valued function defined on Γ, which depends on a parameter N. Consider a Riemann–Hilbert problem for a matrix-valued analytic function R(z, N) that satisfies a jump condition on the contour Γ with the jump matrix V(z, N). Assume that V(z, N) has an asymptotic expansion, as N → ∞, on Γ. An elementary proof is given for the existence of a similar type of asymptotic expansion for the matrix solution R(z, N), as n → ∞, for z ∈ ℂ\Γ. Our method makes use of only complex analysis.

scholarly journals A New Method of Matrix Spectral Factorization

2011 ◽  
Vol 57 (4) ◽  
pp. 2318-2326 ◽  
Author(s):  
G Janashia ◽  
E Lagvilava ◽  
L Ephremidze
Keyword(s):  
New Method ◽  
Spectral Factorization ◽  
Matrix Spectral Factorization
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