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Generalised factorisation for a class of Jones form matrix functions*

Published online by Cambridge University Press:  14 November 2011

M. C. Câmara ,
A. B. Lebre  and
F.-O. Speck
M. C. Câmara
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal
A. B. Lebre
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal
F.-O. Speck
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal
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Synopsis

A systematic approach is proposed for the generalised factorisation of certain non-rational n × n matrix functions. The first main result consists in a transformation of a meromorphic into a generalised factorisation by algebraic means. It closes a gap between the classical Wiener-Hopf procedure and the operator theoretic method of generalised factorisation. Secondly, as examples we consider certain matrix functions of Jones form or of N-part form, which are equivalent to each other, in a sense. The factorisation procedure is complete and explicit, based only on the factorisation of scalar functions, of rational matrix functions and upon linear algebra. Applications in elastodynamic diffraction theory are treated in detail and in a most effective way.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 3 , 1993 , pp. 401 - 422
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Câmara, M. C., Lebre, A. B. and Speck, F.-O.. Meromorphic factorization, partial index estimates and elastodynamic diffraction problems. Math. Nacr. (to appear).Google Scholar
2Chebotarev, G. N.. On the closed form solution of the Riemann boundary value problem for a system of n pairs of functions. Scientific Transactions of the Kazanskij State University V.I. Uljanov-Lenin 116, 4 (1956), 31–58 (in Russian).Google Scholar
3Clancey, K. and Gohberg, I.. Factorization of Matrix Functions and Singular Integral Operators (Basel: Birkhäuser, 1981).CrossRefGoogle Scholar
4Daniele, V. G.. On the factorization of Wiener-Hopf matrices in problems solvable with Hurd's method. I.E.E.E. Trans. Antennas Propag. AP-26 (1978), 614–616.CrossRefGoogle Scholar
5Daniele, V. G.. On the solution of two coupled Wiener-Hopf equations. SIAM J. Appl. Math. 44 (1984), 667–680.CrossRefGoogle Scholar
6Duduchava, R.. Integral Equations with Fixed Singularities (Leipzig: Teubern. 1979).Google Scholar
7Gakhov, F. D.. The Riemann contour problem for a system of n pairs of functions. Uspekhi Mat. Nauk 7, 4 (1952), 3–54.Google Scholar
8Gohberg, I. and Krupnik, N.. Einführung in die Theorie der eindimensionalen singulären Integraloperatoren (Basel: Birkhäuser, 1979).CrossRefGoogle Scholar
9Hurd, R. A.. The explicit factorization of 2 ×2 Wiener-Hopf matrices (Preprint Nr. 1040, Fachbereich Mathematik, Technische Hochschule Darmstadt, 1987).Google Scholar
10Jones, D. S.. Theory of Electromagnetism (London: Pergamon, 1964).Google Scholar
11Jones, D. S.. Commutative Wiener-Hopf factorization of a matrix. Proc. R. Soc. London Ser. A 393 (1984), 185–192.Google Scholar
12Kupradze, V. D.. Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity (Amsterdam: North-Holland, 1979).Google Scholar
13Lebre, A. B.. Operadores de Wiener-Hopf e Factorização de Símbolos (Ph. D. thesis, Instituto Superior Técnico, Lisboa, 1990).Google Scholar
14Lebre, A. B. and dos Santos, A. F.. Generalized factorization for a class of non-rational 2×2 matrix functions. Integral Equations Operator Theory 13 (1990), 671–700.CrossRefGoogle Scholar
15Litvinchuk, G. S. and Spitkovkii, I. M.. Factorization of Measurable Matrix Functions (Basel: Birkhäuser, 1987).Google Scholar
16Meister, E. and Speck, F.-O.. The explicit solution of elastodynamical diffraction problems by symbol factorization. Z. Anal. Anw. 8 (1989), 307–328.CrossRefGoogle Scholar
17Meister, E. and Speck, F.-O.. Wiener-Hopf factorisation of certain non-rational matrix functions in mathematical physics. In The Gohberg Anniversary Collection, Vol. II eds. Dym, H., Goldberg, S., Kaashoek, M. A. and Lancaster, P., Proc. Conf. Calgary 1988 385–394 (Basel: Birkhäuser, 1989).Google Scholar
18Meister, E. and Speck, F.-O.. Modern Wiener-Hopf methods in diffraction theory. In Ordinary and Partial Differential Equations, Vol. 2, eds. Sleeman, B.D. and Jarvis, R. J., Proc. Conf. Dundee, 1988, 130–177 (London: Longman, 1989).Google Scholar
19Mikhlin, S. G. and Prössdorf, S.. Singular Integral Operators (Berlin: Springer, 1986; in German, 1980).Google Scholar
20Noble, B.. Methods Based on the Wiener-Hopf Technique (London: Pergamon, 1958).Google Scholar
21Penzel, F. and Speck, F.-O.. Asymptotic expansion of singular operators on Sobolev spaces (to appear).Google Scholar
22Prossdorf, S.. Some Classes of Singular Equations (Amsterdam: North-Holland, 1978; in German, 1974).Google Scholar
23Prößdorf, S. and Speck, F.-O.. A factorisation procedure for two by two matrix functions on the circle with two rationally independent entries. Proc. R. Soc. Edinburgh Sect. A 115 (1990), 119–138.CrossRefGoogle Scholar
24Rawlins, A. D.. The explicit Wiener-Hopf factorisation of a special matrix. Z. Angew. Math. Mech. 61 (1981), 527–528.CrossRefGoogle Scholar
25Speck, F.-O.. General Wiener-Hopf Factorization Methods (London: Pitman, 1985).Google Scholar
26Talenti, G.. Sulle equazioni integrali di Wiener-Hopf. Boll. Un. Mat. Ital. 7, Suppl. fasc. 1 (1973), 18–118.Google Scholar