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

This article is cited in 15 scientific papers (total in 15 papers)

$J$-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
\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
\jour Russian Math. Surveys
\yr 1973
\vol 28
\issue 1
\pages 69--140

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    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    3. I. V. Kovalishina, “Analytic theory of a class of interpolation problems”, Math. USSR-Izv., 22:3 (1984), 419–463  mathnet  crossref  mathscinet  zmath
    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  crossref
    5. N. K. Al'bov, “On a criterion for solvability of Fredholm equations”, Math. USSR-Sb., 55:1 (1986), 113–119  mathnet  crossref  mathscinet  zmath
    6. L. A. Sakhnovich, “Factorization problems and operator identities”, Russian Math. Surveys, 41:1 (1986), 1–64  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. Rainer Pauli, “Darlington's theorem and complex normalization”, Int J Circ Theor Appl, 17:4 (1989), 429  crossref  mathscinet  isi
    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  crossref
    9. Vladimir Bolotnikov, “On a general moment problem on the half axis”, Linear Algebra and its Applications, 255:1-3 (1997), 57  crossref
    10. Pedro Albgría, Mischa Cotlar, “Generalized Toeplitz Forms and Interpolation Colligations”, Math. Nachr, 190:1 (1998), 5  crossref
    11. N.N. Chernovol, “The degenerate Carathéodory problem and the elementary multiple factor”, Zhurn. matem. fiz., anal., geom., 1:2 (2005), 225–244  mathnet  mathscinet  zmath  elib
    12. Theory Probab. Appl., 51:2 (2007), 342–350  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. A.E.. Choque-Rivero, L.E.. Garza, “Moment perturbation of matrix polynomials”, Integral Transforms and Special Functions, 2014, 1  crossref
    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  mathnet  crossref  isi
    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  mathnet  crossref  crossref  adsnasa  isi  elib
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