Abstract
In the paper [1] a nonstandard approach to the theory of Fourier series was based on the approximation of the group S = {z ∈ ℂ : |z| = 1} by the hyperfinite group G = {z ∈ ℂ : z N = 1} where N is an infinite large natural number. Here we prove the existence of such a hyperfinite approximation for an arbitrary LCA group with a compact open subgroup.
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References
Luxemburg, W.A.J.: ”‘A nonstandard analysis approach to Fourier analysis, contributions to nonstandards analysis”’, Amsterdam, North Holland 1972, p. 16–39.
Gordon, E.I.: ”‘On Fourier transform in nonstandard analysis”’, Izv. Vyssh. Uchebn. Zaved. Math. 1989, N 2 p. 17–25 (in Russian).
Gordon, E.I.: ”‘Hyperfinite approximations of locally compact abelian groups”’, Soviet Math. Dokl. 1991, Vol. 42, N 2 p. 567 – 571.
Gordon, E.I.: ”‘Nonstandard analysis and compact abelian groups”’, Siberian Math. J. 1991, Vol 32. N 2 p. 26 – 40.
Gordon E.J.: ”‘Nonstandard analysis and locally compact abelian groups”’, Acta Applicandae Mathematicae 1991, Vol. 25 p. 221 – 239.
Gordon, E.I.: ”‘Hyperfinite approximations of locally compact groups and some of their applications”’, to appear in the Proceedings of the 5-th Siberian School ”‘Algebra and Analysis’”, Irkutsk, 1991.
Hewitt, E.; Ross, K.: ”‘Abstract Harmonic Analysis”’, 1963, Vol. 1, Springer — Verlag, Berlin-Göttingen-Heidelberg.
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© 1995 Springer Science+Business Media Dordrecht
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Gordon, E.I. (1995). Hyperfinite Approximations of Commutative Topological Groups. In: Albeverio, S.A., Luxemburg, W.A.J., Wolff, M.P.H. (eds) Advances in Analysis, Probability and Mathematical Physics. Mathematics and Its Applications, vol 314. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8451-7_3
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DOI: https://doi.org/10.1007/978-94-015-8451-7_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4481-5
Online ISBN: 978-94-015-8451-7
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