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 Uspekhi Mat. Nauk, 1973, Volume 28, Issue 1(169), Pages 65–130 (Mi umn4835)

$J$-expanding mtrix functions and their role in the analytical theory of electrical circuits

A. V. Efimov, V. P. Potapov

Abstract: Chapter I establishes the essential properties of the $\mathscr A$-matrix of a passive multipole depending on the number of its branches. These properties are based on Langevin's theorem. A classification of the basic objects of investigation:$J$-expanding matrix-functions (class $\mathfrak M$), and also positive matrix functions (class $\mathfrak B$ ), is introduced. Chapter II gives an account of a theory of matrix functions of class $\mathfrak M$. It also investigates the simplest (elementary and primary) matrices of this class. The fact is established that elementary (and primary) factors can be split off from a given matrix of class $\mathfrak M$. In particular, the factorizability of a rational reactive matrix of class $\mathfrak M$ is established.
Chapters III–IV set forth a theory of various subclasses of matrix functions of class $\mathfrak M$: $\mathfrak M_{sl}$, $\mathfrak M_{cgl}$, $\mathfrak M_{lr}$. The realizability of the matrix functions of each of these subclasses as $\mathscr A$-matrices of passive multipoles with the corresponding provision for branches is established.
The fact that they are realizable is proved by the construction of a corresponding multipole.
The last chapter is concerned with a generalization of Darlington's theorem, which leads to a realization of functions of the subclasses $\mathfrak M_{clr}$ and $\mathfrak M_{cglr}$ as $\mathscr A$-matrices or $z$-matrices of dissipative multipoles.

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English version:
Russian Mathematical Surveys, 1973, 28:1, 69–140

Bibliographic databases:

UDC: 519.53+512.83
MSC: 15A48, 15A15, 15A23

Citation: A. V. Efimov, V. P. Potapov, “$J$-expanding mtrix functions and their role in the analytical theory of electrical circuits”, Uspekhi Mat. Nauk, 28:1(169) (1973), 65–130; Russian Math. Surveys, 28:1 (1973), 69–140

Citation in format AMSBIB
\Bibitem{EfiPot73} \by A.~V.~Efimov, V.~P.~Potapov \paper $J$-expanding mtrix functions and their role in the analytical theory of electrical circuits \jour Uspekhi Mat. Nauk \yr 1973 \vol 28 \issue 1(169) \pages 65--130 \mathnet{http://mi.mathnet.ru/umn4835} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=394287} \zmath{https://zbmath.org/?q=an:0268.94009|0285.94009} \transl \jour Russian Math. Surveys \yr 1973 \vol 28 \issue 1 \pages 69--140 \crossref{https://doi.org/10.1070/RM1973v028n01ABEH001397} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. Z. Arov, “Darlington realization of matrix-valued functions”, Math. USSR-Izv., 7:6 (1973), 1295–1326
2. S. A. Orlov, “Nested matrix disks analytically depending parameter, and theorems on the invariance radii of limiting disks”, Math. USSR-Izv., 10:3 (1976), 565–613
3. I. V. Kovalishina, “Analytic theory of a class of interpolation problems”, Math. USSR-Izv., 22:3 (1984), 419–463
4. Elsa Cortina, “j-Expansive matrix-valued functions and Darlington realization of transfer-scattering matrices”, Journal of Mathematical Analysis and Applications, 92:2 (1983), 435
5. N. K. Al'bov, “On a criterion for solvability of Fredholm equations”, Math. USSR-Sb., 55:1 (1986), 113–119
6. L. A. Sakhnovich, “Factorization problems and operator identities”, Russian Math. Surveys, 41:1 (1986), 1–64
7. Rainer Pauli, “Darlington's theorem and complex normalization”, Int J Circ Theor Appl, 17:4 (1989), 429
8. AndréC.M Ran, Leiba Rodman, “Laurent interpolation for rational matrix functions and a local factorization principle”, Journal of Mathematical Analysis and Applications, 164:2 (1992), 524
9. Vladimir Bolotnikov, “On a general moment problem on the half axis”, Linear Algebra and its Applications, 255:1-3 (1997), 57
10. Pedro Albgría, Mischa Cotlar, “Generalized Toeplitz Forms and Interpolation Colligations”, Math. Nachr, 190:1 (1998), 5
11. N.N. Chernovol, “The degenerate Carathéodory problem and the elementary multiple factor”, Zhurn. matem. fiz., anal., geom., 1:2 (2005), 225–244
12. Theory Probab. Appl., 51:2 (2007), 342–350
13. A.E.. Choque-Rivero, L.E.. Garza, “Moment perturbation of matrix polynomials”, Integral Transforms and Special Functions, 2014, 1
14. A. E. Choke Rivero, L. E. Garza Gaona, “Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices”, Russian Math. (Iz. VUZ), 61:12 (2017), 57–69
15. Yu. M. Dyukarev, “The zeros of determinants of matrix-valued polynomials that are orthonormal on a semi-infinite or finite interval”, Sb. Math., 209:12 (2018), 1745–1755
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