What is the definition of an antilimit?

Question

I've seen some references to antilimits in the numerical analysis literature, but no definition of the term. The impression I get is that in specific contexts where every sequence $x_0,x_1,x_2,\dots$ under consideration has a unique extrapolation backward to $x_{-1},x_{-2},x_{-3},\dots$, and this extrapolated sequence converges to a limit $L$, we say that $L$ is an antilimit of the original sequence.

Is that all there is to it?

Can anyone provide information on contexts in which the concept of antilimits is useful?

Answer 1

Antilimits are used to apply the methods of sequence transformations to divergent series. There is no unique definition, but typically if the sum $\sum_{n}a_n(x)$ converges to some function $f(x)$ for $|x|<\rho$, and this function can be continued analytically for $|x|>\rho$, then $f(x)$ is called the limit of the divergent series for $|x|<\rho$ and the antilimit for $|x|>\rho$.

A classic application (from Christopher Small's Expansions and Asymptotics for Statistics) is the proof of the identity

$$\frac{1}{1-x}+\frac{1}{1+x^2}-\frac{x}{1-x^2}=\frac{2}{1-x^4}$$

by sequence transformations:

$$\frac{1}{1-x}+\frac{1}{1+x^2}-\frac{x}{1-x^2}=(1+x+x^2+\cdots)+$$ $$(1-x^2+x^4-\cdots)-(x+x^3+x^5+\cdots)$$ $$=2+2x^4+2x^8+\cdots=\frac{2}{1-x^4}.$$

Although the sums of the series only have a limit for $|x|<1$, the proof of the identity remains valid for $|x|>1$ if the sums are interpreted as antilimits.

Many more applications of antilimits in sequence transformations can be found in Avram Sidi's Practical Extrapolation Methods.

Answer 2

I'd prefer to go plural on this by saying that antilimits are numbers or expressions that can emerge when techniques that, for example, are employed to increase the speed of convergence of series are used outside convergence limits. I'd never before seen negative subscripts, a concept whose utility in interpolation is intuitively unmistakeable.