Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Published electronically: August 25, 1999
Supplement: Additional information related to this article.
MathSciNet review: 1659831
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let

\begin{displaymath}G(x)=\left[\begin{matrix}e^{i\lambda x}I_m & 0\\ c_{-1}e^{-i\nu x}+c_0+c_1 e^{i\alpha x} & e^{-i\lambda x}I_m \end{matrix}\right],\end{displaymath}

where $c_j\in\mathbb{C}^{m\times m}$, $\alpha,\nu>0$ and $\alpha+\nu=\lambda$. For rational $\alpha/\nu$ such matrices $G$ are periodic, and their Wiener-Hopf factorization with respect to the real line $\mathbb{R}$ always exists and can be constructed explicitly. For irrational $\alpha/\nu$, a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible $c_0$ and commuting $c_1c_0^{-1}$, $c_{-1}c_0^{-1}$ was disposed of earlier-it was discovered that an almost periodic factorization of such matrices $G$ does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when $c_0$ is not invertible but the $c_j$ commute pairwise ($j=0,\pm 1$). The complete description is obtained when $m\leq 3$; for an arbitrary $m$, certain conditions are imposed on the Jordan structure of $c_j$. Difficulties arising for $m=4$ 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 $G$ in numerous particular cases. These factorizations were obtained using Maple; the code is available from the authors upon request.

References [Enhancements On Off] (What's this?)

  • 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 $\psi (x+\alpha )-b(x) \psi (x)=g(x)$, 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

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 47A68, 47-04, 42A75

Retrieve articles in all journals with MSC (1991): 47A68, 47-04, 42A75

Additional Information

Ilya M. Spitkovsky

Darryl Yong

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

American Mathematical Society