Almost periodic factorization of certain

block triangular matrix functions

Authors:
Ilya M. Spitkovsky and Darryl Yong

Journal:
Math. Comp. **69** (2000), 1053-1070

MSC (1991):
Primary 47A68, 47-04, 42A75

DOI:
https://doi.org/10.1090/S0025-5718-99-01161-8

Published electronically:
August 25, 1999

Supplement:
Additional information related to this article.

MathSciNet review:
1659831

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Abstract: Let

where , and . For rational such matrices are periodic, and their Wiener-Hopf factorization with respect to the real line always exists and can be constructed explicitly. For irrational , a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible and commuting , was disposed of earlier-it was discovered that an almost periodic factorization of such matrices does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when is not invertible but the commute pairwise (). The complete description is obtained when ; for an arbitrary , certain conditions are imposed on the Jordan structure of . Difficulties arising for are explained, and a classification of both solved and unsolved cases is given. The main result of the paper (existence criterion) is theoretical; however, a significant part of its proof is a constructive factorization of in numerous particular cases. These factorizations were obtained using Maple; the code is available from the authors upon request.

**1.**M. Bakonyi, L. Rodman, I. Spitkovsky, and H. Woerdeman,*Positive extensions of matrix functions of two variables with support in an infinite band*, C. R. Acad. Sci. Paris Sér. I Math.**323**(1996), no. 8, 859-863. MR**97i:47023****2.**M. A. Bastos, Yu. I. Karlovich, I. M. Spitkovsky, and P. M. Tishin,*On a new algorithm for almost periodic factorization*, Operator Theory: Advances and Applications**103**(1998), 53-74. CMP**98:16****3.**C. Corduneanu,*Almost periodic functions*, J. Wiley & Sons, 1968. MR**58:2006****4.**N. K. Karapetjanc and S. G. Samko,*The functional equation*, Izv. Akad. Nauk Armjan. SSR. Ser. Mat.**5**(1970), no. 5, 441-448. MR**44:2101****5.**Yu. I. Karlovich,*On the Haseman problem*, Demonstratio Math.**26**(1993), 581-595. MR**95a:47048****6.**Yu. I. Karlovich and I. M. Spitkovsky,*Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type*, Mathematics of the USSR, Izvestiya**34**(1990), 281-316. MR**90f:47034****7.**-,*(Semi)-Fredholmness of convolution operators on the spaces of Bessel potentials*, Operator Theory: Advances and Applications**71**(1994), 122-152. MR**95h:47034****8.**-,*Almost periodic factorization: An analogue of Chebotarev's algorithm*, Contemporary Math.**189**(1995), 327-352. MR**96h:47024****9.**-,*Factorization of almost periodic matrix functions*, J. Math. Anal. Appl.**193**(1995), 209-232. MR**96m:47047****10.**-,*Semi-Fredholm properties of certain singular integral operators*, Operator Theory: Advances and Applications**90**(1996), 264-287. MR**97k:47046****11.**B. M. Levitan,*Almost periodic functions*, GITTL, Moscow, 1953 (in Russian). MR**15:700a****12.**B. M. Levitan and V. V. Zhikov,*Almost periodic functions and differential equations*, Cambridge University Press, 1982. MR**84g:34004****13.**G. S. Litvinchuk and I. M. Spitkovsky,*Factorization of measurable matrix functions*, Birkhäuser Verlag, Basel and Boston, 1987. MR**90g:47030****14.**Yu. Lyubarskii and I. Spitkovsky,*Sampling and interpolating for a lacunary spectrum*, Royal Society of Edinburgh, Proceedings**126A**(1996), 77-87. MR**97b:41004****15.**L. Rodman, I. M. Spitkovsky, and H. J. Woerdeman,*Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions*, Trans. Amer. Math. Soc.**350**(1998), 2185-2227. MR**98h:47023****16.**I. M. Spitkovsky,*On the factorization of almost periodic matrix functions*, Math. Notes**45**(1989), no. 5-6, 482-488. MR**90k:47033****17.**I. M. Spitkovsky and H. J. Woerdeman,*The Carathèodory-Toeplitz problem for almost periodic functions*, J. Functional Analysis**115**(1993), no. 2, 281-293. MR**94f:47020**

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Additional Information

**Ilya M. Spitkovsky**

Email:
ilya@math.wm.edu

**Darryl Yong**

Email:
dyong@u.washington.edu

DOI:
https://doi.org/10.1090/S0025-5718-99-01161-8

Keywords:
Almost periodic matrix functions,
factorization,
explicit computation

Received by editor(s):
March 12, 1997

Received by editor(s) in revised form:
September 18, 1998

Published electronically:
August 25, 1999

Additional Notes:
The first author’s research was partially supported by NSF Grant DMS-9800704

The second author’s research was started during a Research Experience for Undergraduates sponsored by the NSF at the College of William and Mary during the summer of 1995.

Article copyright:
© Copyright 2000
American Mathematical Society