Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Canonical factorization of continuous functions on the -torus

Author(s): Torsten Ehrhardt; Cornelis V. M. van der Mee
Journal: Proc. Amer. Math. Soc. 131 (2003), 801-813.
MSC (1991): Primary 46J10; Secondary 43A20
Posted: July 26, 2002
MathSciNet review: 1937418
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Abstract: In this article we study the canonical factorization of continuous complex-valued functions on the -dimensional torus belonging to a weighted Wiener algebra with respect to a linear order on the -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:

1.: R. Arens, The group of invertible elements of a commutative Banach algebra, Studia Math. Zeszyt (Ser. Specjalna) 1 (1963), 21-23. MR 26:4198
2.: A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Springer, Berlin, 1990. MR 92e:47001
3.: L. Cerlienco and M. Mureddu, Rappresentazione matriciale degli ordini l.c. su ${\mathbb{R}}^n$ e su ${\mathbb{N}}^n$ , Rend. Sem. Fac. Sc. Univ. Cagliari 66 (1996), 49-68. MR 98c:06025
4.: R.G. Douglas, Banach Algebra Techniques in Operator Theory, Graduate Texts in Mathematics, vol. 179, Springer, Berlin, 1998. MR 99c:47001
5.: J. Erdos, On the structure of ordered real vector spaces, Publ. Math. Debrecen 4 (1956), 334-343. MR 18:137b
6.: I. Gelfand, D. Raikov, and G. Shilov, Commutative Normed Rings, AMS Chelsea Publ., Providence, RI, 1964. MR 34:4940
7.: I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Vol. II, Birkhäuser, vol. OT 63, Basel-Boston, 1993. MR 95a:47001
8.: T.N.T. Goodman, C.A. Micchelli, G. Rodriguez, and S. Seatzu, On the Cholesky factorization of the Gram matrix of multivariate functions, SIAM J. Matrix Anal. Appl. 22 (2000), 501-526. MR 2001k:15020
9.: 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. MR 20:4155; MR 31:562
10.: I. N. Herstein, Topics in Algebra, Xerox College Publ., Waltham, Mass., 1964. MR 50:9456
11.: E. Hille and R.S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. Coll. Publ., vol. 31, Amer. Math. Soc., Providence, RI, 1957. MR 19:664d
12.: J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer, Berlin, 1993. MR 95m:90001
13.: Sze-tsen Hu, Homotopy Theory, Pure and Applied Mathematics, vol. 8, New York, Academic Press, 1959. MR 21:5186
14.: M.G. Krein, Integral equations on the half-line with a kernel depending on the difference of the arguments, AMS Transl. Ser. 2, 22 (1962), 163-288; transl. from Uspehi Mat. Nauk. 13 (1958), 3-120. MR 21:1507
15.: 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.
16.: M.A. Naimark, Normed algebras, Wolters-Noordhoff Publ., Groningen, 1972. MR 55:11042
17.: H.L. Royden, Function algebras, Bull. Amer. Math. Soc. 69 (1963), 281-298. MR 26:6817
18.: W. Rudin, Fourier Analysis on Groups, John Wiley, New York, 1962. MR 27:2808
19.: I.B. Simonenko, On multidimensional discrete convolutions, Matem. Issled. Kishinev 3(1) (1968), 108-122 [Russian]. MR 41:2412
20.: J.L. Taylor, Banach algebras and topology, In: Algebras in Analysis, Academic Press, London 1975, pp. 118-186. MR 54:5837

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