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Some remarks on formal power series and formal Laurent series

Dariusz Bugajewski and Xiao-Xiong Gan
From the journal Mathematica Slovaca


In this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X.—BUGAJEWSKI, D.:On formal Laurent series, Bull. Braz. Math. Soc. 42 (2011), 415–437], is not continuous. We also show some applications of fixed point theorems to some nonlinear mappings defined on the space of formal power series or on the space of formal Laurent series. Finally, in the second part of the paper, we propose another approach to study of dot product and multiplication of formal Laurent series, in particular establishing integral representation of dot product and convolution representation of multiplication.

(Communicated by Stanisława Kanas)


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Received: 2014-12-14
Accepted: 2015-5-17
Published Online: 2017-6-5
Published in Print: 2017-6-27

© 2017 Mathematical Institute Slovak Academy of Sciences