(Near-)inverses of sequences

Abstract

We introduce the notion of a near-inverse of a non-decreasing sequence of positive integers; near-inverses are intended to assume the role of inverses in cases when the latter cannot exist. We prove that the near-inverse of such a sequence is unique; moreover, the relation of being near-inverses of each other is symmetric, i.e. if sequence g is the near-inverse of sequence f, then f is the near-inverse of g. There is a connection, by approximations, between near-inverses of sequences and inverses of continuous strictly increasing real-valued functions which can be exploited to derive simple expressions for near-inverses.

Keywords:

• Integer sequence
• Inverse function
• Solid code

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada through Grants OGP0000243 and OGP220259.