modulus of convergence

In real analysis, a branch of mathematics, a modulus of convergence is a function that tells how quickly a convergent sequence converges. These moduli are often employed in the study of computable analysis and constructive mathematics.

If a sequence of real numbers $x_i$ converges to a real number $x$ , then by definition, for every real $\varepsilon > 0$ there is a natural number $N$ such that if $i > N$ then $\left|x - x_i\right| < \varepsilon$ . A modulus of convergence is essentially a function that, given $\varepsilon$ , returns a corresponding value of $N$ .

Definition

Suppose that $x_i$ is a convergent sequence of real numbers with limit $x$ . There are two ways of defining a modulus of convergence as a function from natural numbers to natural numbers:

As a function $f$ such that for all $n$ , if $i > f(n)$ then $\left|x - x_i\right| < 1/n$ .
As a function $g$ such that for all $n$ , if $i \geq j > g(n)$ then $\left|x_i - x_j\right| < 1/n$ .

The latter definition is often employed in constructive settings, where the limit $x$ may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces $1/n$ with $2^{-n}$ .

References

Klaus Weihrauch (2000), Computable Analysis.

Category:Constructivism (mathematics)

Category:Computable analysis

Category:Real analysis

modulus of convergence

Definition

See also

References