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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access April 15, 2016

On the Bézout equation in the ring of periodic distributions

  • Rudolf Rupp and Amol Sasane

Abstract

A corona type theorem is given for the ring D'A(Rd) of periodic distributions in Rd in terms of the sequence of Fourier coefficients of these distributions,which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of D'A(Rd) are both equal to 1.

References

[1] H. Bass. K-theory and stable algebra. Inst. Hautes Études Sci. Publ. Math., No. 22, 5-60, 1964. 10.1007/BF02684689Search in Google Scholar

[2] N.K. Bose. Applied Multidimensional Systems Theory. Reidel, Dordrecht, 1984. 10.1007/978-94-009-5225-6Search in Google Scholar

[3] W.F. Donoghue, Jr., Distributions and Fourier Transforms. Pure and Applied Mathematics 32, Academic Press, New York and London, 1969. Search in Google Scholar

[4] J.J. Duistermaat and J.A.C. Kolk. Distributions. Theory and Applications. Birkhäuser, Boston, MA, 2010. 10.1007/978-0-8176-4675-2Search in Google Scholar

[5] L. Hörmander. Generators for some rings of analytic functions. Bull. Am. Math. Soc., 73:943–949, 1967. 10.1090/S0002-9904-1967-11860-3Search in Google Scholar

[6] L. Hörmander. The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Second edition. Springer Study Edition, Springer-Verlag, Berlin, 1990. Search in Google Scholar

[7] T.Y. Lam. A first course in noncommutative rings. Second edition, Graduate Texts in Mathematics Volume 131, Springer- Verlag, New York, 2001. 10.1007/978-1-4419-8616-0Search in Google Scholar

[8] M.A. Rieffel. Dimension and stable rank in the K-theory of C*-algebras. Proceedings of the London Mathematical Society, (3), 46:301-333, no. 2, 1983. 10.1112/plms/s3-46.2.301Search in Google Scholar

[9] S. Maad Sasane and A.J. Sasane. Generators for rings of compactly supported distributions. Integral Equations Operator Theory, 69:63-71, no. 1, 2011. 10.1007/s00020-010-1842-3Search in Google Scholar

[10] F. Trèves. Topological Vector Spaces, Distributions and Kernels. Unabridged republication of the 1967 original. Dover Publications, Mineola, NY, 2006. Search in Google Scholar

Received: 2015-6-4
Accepted: 2015-10-18
Published Online: 2016-4-15

© 2016 Rudolf Rupp and Amol Sasane

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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