# Moebius transforms, continued fractions and Padé approximants

## Abstract

In this chapter we shall introduce some basic tools and definitions. Some of the definitions will be slight modifications of the classical definitions. These modifications are needed since we shall treat formal Laurent series rather than formal power series which causes some problems for the definition of the quotient and product of such series. Moebius transforms are simple linear fractional transforms. They are important because continued fractions, to be treated in the third subsection, can be defined as a sequence of Moebius transforms of a special type. Another tool we shall introduce in the second subsection are flow graphs. These are extensively used in the engineering literature to give a graphical representation of an algorithm or an analog network realizing this algorithm. Some of the flow graphs we shall give will probably look very familiar to readers who know the ladder or lattice realizations of prediction filters. It may help them to understand the theory and it will be advantageous for others to have a clear and concise representation of a certain algorithm. To introduce our formal concepts of different types of Padé approximants, which is done in the last subsection, we need some definitions and notations to work with formal series. These are given in subsection 4.

## Keywords

Sink Node Continue Fraction Formal Power Series Formal Series Laurent Series## Preview

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