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Motion Groups and Absolutely Convergent Fourier Transforms

Part of: Approximations and expansions Lie groups Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Garthi Gaudry  and
Rita Pini
Garthi Gaudry
Affiliation:
School of Mathematical SciencesThe Flinders University of South AustraliaBedford Park S.A. 5042, Australia
Rita Pini
Affiliation:
Via Lattanzio 16 20127 Milano, Italy
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According to an extension of a classical theorem of Bernstein, due to C. Herz, a function on Rn belonging to a Besov space of appropriate order has an absolutely convergent Fourier transform. We establish extensions of this result to Cartan motion groups, for Besov spaces defined with respect to both isotropic and non-isotropic differences.

MSC classification

Secondary: 22E30: Analysis on real and complex Lie groups 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities) 43A50: Convergence of Fourier series and of inverse transforms 43A80: Analysis on other specific Lie groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 3 , December 1987 , pp. 385 - 397
Copyright
Copyright © Australian Mathematical Society 1987

References

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[3]Herz, C. S., ‘Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms’, J. Math. Mech. 18 (1968), 283–323.Google Scholar
[4]Inglis, I. R., ‘Bernstein's theorem and the Fourier algebra of the Heisenberg group’, Boll. Un. Math. hal. A (6) 2 (1983), 39–46.Google Scholar