Almost periodic factorization of certain block triangular matrix functions
Authors:
Ilya M. Spitkovsky and Darryl Yong
Journal:
Math. Comp. 69 (2000), 10531070
MSC (1991):
Primary 47A68, 4704, 42A75
Published electronically:
August 25, 1999
Supplement:
Additional information related to this article.
MathSciNet review:
1659831
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let where , and . For rational such matrices are periodic, and their WienerHopf 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 earlierit 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.
Mihály
Bakonyi, Leiba
Rodman, Ilya
M. Spitkovsky, and Hugo
J. 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
(English, with English and French summaries). MR 1414548
(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), 5374. CMP 98:16
 3.
C.
Corduneanu, Almost periodic functions, Interscience Publishers
[John Wiley & Sons], New YorkLondonSydney, 1968. With the
collaboration of N. Gheorghiu and V. Barbu; Translated from the Romanian by
Gitta Bernstein and Eugene Tomer; Interscience Tracts in Pure and Applied
Mathematics, No. 22. MR 0481915
(58 #2006)
 4.
N.
K. Karapetjanc and S.
G. Samko, The functional equation
𝜓(𝑥+𝛼)𝑏(𝑥)𝜓(𝑥)=𝑔(𝑥),
Izv. Akad. Nauk Armjan. SSR Ser. Mat. 5 (1970),
no. 5, 441–448 (Russian, with Armenian and English summaries).
MR
0284877 (44 #2101)
 5.
Yu.
I. Karlovich, On the Haseman problem, Demonstratio Math.
26 (1993), no. 34, 581–595 (1994). MR 1265822
(95a:47048)
 6.
Yu.
I. Karlovich and I.
M. Spitkovskiĭ, Factorization of almost periodic matrix
functions and the Noether theory of certain classes of equations of
convolution type, Izv. Akad. Nauk SSSR Ser. Mat. 53
(1989), no. 2, 276–308 (Russian); English transl., Math.
USSRIzv. 34 (1990), no. 2, 281–316. MR 998297
(90f:47034)
 7.
Yuri
Karlovich and Ilya
Spitkovsky, (Semi)Fredholmness of convolution operators on the
spaces of Bessel potentials, Toeplitz operators and related topics
(Santa Cruz, CA, 1992) Oper. Theory Adv. Appl., vol. 71,
Birkhäuser, Basel, 1994, pp. 122–152. MR 1300217
(95h:47034)
 8.
Yuri
Karlovich and Ilya
Spitkovsky, Almost periodic factorization: an analogue of
Chebotarev’s algorithm, Harmonic analysis and operator theory
(Caracas, 1994) Contemp. Math., vol. 189, Amer. Math. Soc.,
Providence, RI, 1995, pp. 327–352. MR 1347021
(96h:47024), http://dx.doi.org/10.1090/conm/189/02271
 9.
Yuri
Karlovich and Ilya
Spitkovsky, Factorization of almost periodic matrix functions,
J. Math. Anal. Appl. 193 (1995), no. 1,
209–232. MR 1338508
(96m:47047), http://dx.doi.org/10.1006/jmaa.1995.1230
 10.
Yuri
I. Karlovich and Ilya
M. Spitkovsky, SemiFredholm properties of certain singular
integral operators, Singular integral operators and related topics
(Tel Aviv, 1995) Oper. Theory Adv. Appl., vol. 90, Birkhäuser,
Basel, 1996, pp. 264–287. MR 1413556
(97k:47046)
 11.
B.
M. Levitan, Počtiperiodičeskie funkcii,
Gosudarstv. Izdat. Tehn.Teor. Lit., Moscow, 1953 (Russian). MR 0060629
(15,700a)
 12.
B.
M. Levitan and V.
V. Zhikov, Almost periodic functions and differential
equations, Cambridge University Press, Cambridge, 1982. Translated
from the Russian by L. W. Longdon. MR 690064
(84g:34004)
 13.
Georgii
S. Litvinchuk and Ilia
M. Spitkovskii, Factorization of measurable matrix functions,
Operator Theory: Advances and Applications, vol. 25, Birkhäuser
Verlag, Basel, 1987. Translated from the Russian by Bernd Luderer; With a
foreword by Bernd Silbermann. MR 1015716
(90g:47030)
 14.
Yu.
Lyubarskii and I.
Spitkovsky, Sampling and interpolation for a lacunary
spectrum, Proc. Roy. Soc. Edinburgh Sect. A 126
(1996), no. 1, 77–87. MR 1378833
(97b:41004), http://dx.doi.org/10.1017/S0308210500030602
 15.
Leiba
Rodman, Ilya
M. Spitkovsky, and Hugo
J. Woerdeman, CarathéodoryToeplitz and
Nehari problems for matrix valued almost periodic functions, Trans. Amer. Math. Soc. 350 (1998), no. 6, 2185–2227. MR 1422908
(98h:47023), http://dx.doi.org/10.1090/S0002994798019370
 16.
I.
M. Spitkovskiĭ, Factorization of almostperiodic matrix
functions, Mat. Zametki 45 (1989), no. 6,
74–82, 111 (Russian); English transl., Math. Notes
45 (1989), no. 56, 482–488. MR 1019039
(90k:47033), http://dx.doi.org/10.1007/BF01158238
 17.
Ilya
M. Spitkovsky and Hugo
J. Woerdeman, The CarathéodoryToeplitz problem for almost
periodic functions, J. Funct. Anal. 115 (1993),
no. 2, 281–293. MR 1234392
(94f:47020), http://dx.doi.org/10.1006/jfan.1993.1091
 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, 859863. 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), 5374. 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, 441448. MR 44:2101
 5.
 Yu. I. Karlovich, On the Haseman problem, Demonstratio Math. 26 (1993), 581595. MR 95a:47048
 6.
 Yu. I. Karlovich and I. M. Spitkovsky, Factorization of almost periodic matrixvalued functions and the Noether theory for certain classes of equations of convolution type, Mathematics of the USSR, Izvestiya 34 (1990), 281316. MR 90f:47034
 7.
 , (Semi)Fredholmness of convolution operators on the spaces of Bessel potentials, Operator Theory: Advances and Applications 71 (1994), 122152. MR 95h:47034
 8.
 , Almost periodic factorization: An analogue of Chebotarev's algorithm, Contemporary Math. 189 (1995), 327352. MR 96h:47024
 9.
 , Factorization of almost periodic matrix functions, J. Math. Anal. Appl. 193 (1995), 209232. MR 96m:47047
 10.
 , SemiFredholm properties of certain singular integral operators, Operator Theory: Advances and Applications 90 (1996), 264287. 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), 7787. MR 97b:41004
 15.
 L. Rodman, I. M. Spitkovsky, and H. J. Woerdeman, CarathéodoryToeplitz and Nehari problems for matrix valued almost periodic functions, Trans. Amer. Math. Soc. 350 (1998), 21852227. MR 98h:47023
 16.
 I. M. Spitkovsky, On the factorization of almost periodic matrix functions, Math. Notes 45 (1989), no. 56, 482488. MR 90k:47033
 17.
 I. M. Spitkovsky and H. J. Woerdeman, The CarathèodoryToeplitz problem for almost periodic functions, J. Functional Analysis 115 (1993), no. 2, 281293. MR 94f:47020
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
47A68,
4704,
42A75
Retrieve articles in all journals
with MSC (1991):
47A68,
4704,
42A75
Additional Information
Ilya M. Spitkovsky
Email:
ilya@math.wm.edu
Darryl Yong
Email:
dyong@u.washington.edu
DOI:
http://dx.doi.org/10.1090/S0025571899011618
PII:
S 00255718(99)011618
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 DMS9800704
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
