Abstract
In a recent paper on nonlocal expansions necessary and sufficient conditions are given under whichf −1 has a generalized power series expansion, whenf is an invertible locally Lipschitz map between certain general subsets of a complex Banach space. Here we establish the validity of a conceptually interesting algorithm for obtaining the expansion.
Basically, we show that a certain contraction mapping iteration generates iteratesℐ 1,ℐ 2,... such that eachℐ k yields all of the terms of the generalized power series expansion off −1 up to order (k + 1), assuming merely that the expansion off −1 exists. An earlier different result along related lines is mentioned.
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Sandberg, I.W. Iteration and functional expansions. Circuits Systems and Signal Process 3, 409–417 (1984). https://doi.org/10.1007/BF01599168
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DOI: https://doi.org/10.1007/BF01599168