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On certain non-archimedean functions analogous to complex analytic functions

Published online by Cambridge University Press:  17 April 2009

Kurt Mahler
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Let Qp and K be the rational p–adic field and an algebraic extension of Qp of finite degree, respectively, and let I and Ik be the subsets of Qp and of K consisting of the p–adic integers of these fields.

It is known that the continuous functions f: IQp can be written as

where this series converges uniformly on I, and that such continuous functions need not be differentiable at any point. We here study continuous functions F: IKK which for all X on IK are the sum of a uniformly convergent series

It is proved that such functions F(X) have at every point of IK derivatives of all orders. In the special case when K is totally ramified, they cannot in general be developed into power series that converge everywhere on IK, but this is possible when K is not totally ramified.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Mahler, K., “An interpolation series for continuous functions of a p–adic variable”, J. reine angew. Math. 199 (1958), 2334.CrossRefGoogle Scholar