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Characterization of inverse-closed Beurling classes

Published online by Cambridge University Press:  04 October 2011

Jamil A. Siddiqi  and
Mostefa Ider
Jamil A. Siddiqi
Affiliation:
Département de Mathématiques, Université Laval, Québec G1K 7P4, Canada
Mostefa Ider
Affiliation:
Département de Mathématiques, Université Laval, Québec G1K 7P4, Canada
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Extract

In [1], J. Bruna studied the Beurling classes EM(I) of infinitely differentiable functions f defined on an interval I such that for every positive ε, there exists a constant Cε > 0 with the property that

where M = {Mn} is a given sequence of positive numbers. With the hypothesis that the class EM(ℝ) is differentiable, he proved that it is inverse-closed (in the sense that if fεEM(ℝ) and if f is bounded away from zero on ℝ, then its inverse f-1 lies in EM(ℝ)) if and only if the associated sequence A = {An = (Mn/n!)1/n} is almost increasing (i.e. Am ≤ KAn for all m ≤ n, where K > 0 is a constant independent of m and n).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 495 - 501
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

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