Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Canonical factorization of continuous functions on the $d$-torus

Authors: Torsten Ehrhardt and Cornelis V. M. van der Mee
Journal: Proc. Amer. Math. Soc. 131 (2003), 801-813
MSC (1991): Primary 46J10; Secondary 43A20
Published electronically: July 26, 2002
MathSciNet review: 1937418
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we study the canonical factorization of continuous complex-valued functions on the $d$-dimensional torus belonging to a weighted Wiener algebra with respect to a linear order on the $d$-tuples of integers. It is proved that a function has a canonical factorization in this algebra if and only if it has a logarithm belonging to this algebra. A second characterization is given in terms of winding numbers. Moreover, the maximal ideal spaces of the relevant Banach algebras are identified.

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

  • R. Arens, The group of invertible elements of a commutative Banach algebra, Studia Math. (Ser. Specjalna) Zeszyt 1 (1963), 21–23. MR 0146678
  • Albrecht Böttcher and Bernd Silbermann, Analysis of Toeplitz operators, Springer-Verlag, Berlin, 1990. MR 1071374
  • L. Cerlienco and M. Mureddu, Matrix representation of compatible linear orders on ${\bf R}^n$ and ${\bf N}^n$, Rend. Sem. Fac. Sci. Univ. Cagliari 66 (1996), no. 1, 49–68 (Italian, with English summary). MR 1445221
  • Ronald G. Douglas, Banach algebra techniques in operator theory, 2nd ed., Graduate Texts in Mathematics, vol. 179, Springer-Verlag, New York, 1998. MR 1634900
  • J. Erdős, On the structure of ordered real vector spaces, Publ. Math. Debrecen 4 (1956), 334-343.
  • I. Gelfand, D. Raikov, and G. Shilov, Commutative normed rings, Chelsea Publishing Co., New York, 1964. Translated from the Russian, with a supplementary chapter. MR 0205105
  • Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek, Classes of linear operators. Vol. II, Operator Theory: Advances and Applications, vol. 63, Birkhäuser Verlag, Basel, 1993. MR 1246332
  • Tim N. T. Goodman, Charles A. Micchelli, Giuseppe Rodriguez, and Sebastiano Seatzu, On the Cholesky factorization of the Gram matrix of multivariate functions, SIAM J. Matrix Anal. Appl. 22 (2000), no. 2, 501–526. MR 1780197, DOI
  • H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165-202; part II: Acta Math. 106 (1961), 175-213. ;
  • I. N. Herstein, Topics in algebra, 2nd ed., Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. MR 0356988
  • E. Hille and R.S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. Coll. Publ., vol. 31, Amer. Math. Soc., Providence, RI, 1957.
  • Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal, Convex analysis and minimization algorithms. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305, Springer-Verlag, Berlin, 1993. Fundamentals. MR 1261420
  • Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
  • M. G. Kreĭn, Integral equations on the half-line with a kernel depending on the difference of the arguments, Uspehi Mat. Nauk 13 (1958), no. 5 (83), 3–120 (Russian). MR 0102721
  • C.V.M. van der Mee, G. Rodriguez, and S. Seatzu, Spectral factorization of bi-infinite multi-index block Toeplitz matrices, Linear Algebra and its Applications 343-344 (2002), 355–380.
  • M. A. Naĭmark, Normed algebras, 3rd ed., Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the second Russian edition by Leo F. Boron; Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics. MR 0438123
  • H. L. Royden, Function algebras, Bull. Amer. Math. Soc. 69 (1963), 281–298. MR 149327, DOI
  • Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
  • I. B. Simonenko, Multidimensional discrete convolutions, Mat. Issled. 3 (1968), no. vyp. 1 (7), 108–122 (Russian). MR 0257763
  • J. L. Taylor, Banach algebras and topology, Algebras in analysis (Proc. Instructional Conf. and NATO Advanced Study Inst., Birmingham, 1973) Academic Press, London, 1975, pp. 118–186. MR 0417789

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46J10, 43A20

Retrieve articles in all journals with MSC (1991): 46J10, 43A20

Additional Information

Torsten Ehrhardt
Affiliation: Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany

Cornelis V. M. van der Mee
Affiliation: Dipartimento di Matematica, Università di Cagliari, via Ospedale 72, 09124 Cagliari, Italy

Keywords: Canonical factorization, Wiener algebra, maximal ideal space
Received by editor(s): July 18, 2001
Received by editor(s) in revised form: October 12, 2001
Published electronically: July 26, 2002
Additional Notes: This research was partially supported by INdAM-GNCS and MURST
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society