xbeta
Uniform convergence

Loosely speaking, a sequence of functions ${f_{n}}$ converges uniformly to a function $f$ if $f_{n} (x) \to f (x)$ pointwise for all $x$ at a rate that is independent of $x$ .

Definition

We say that a sequence of real-valued functions ${f_{n}}$ with $f_{n} : A \to ℝ$ converges uniformly to $f : A \to ℝ$ if for every $ε > 0$ , there exists a number $N$ such that for all $x \in A$ and all $n \geq N$ , $∣ f_{n} (x) - f (x) ∣ < ε$ .

Alternatively, ${f_{n}}$ is uniformly convergent to $f$ if and only if

\sup_{x \in A} ∣ f_{n} (x) - f (x) ∣ \to 0 .

This alternative form is commonly used as the definition.

Applications

If ${f_{n}}$ is a sequence of continuous functions which converges uniformly towards the function $f$ on $A$ , then $f$ is also continuous on $A$ .

Related Concepts

Stochastic equicontinuity?
Uniform law of large numbers?

category: Mathematics, Econometrics

Created on April 23, 2009 16:55:59 by Jason Blevins (207.192.72.34)

xbeta Uniform convergence

Definition

Applications

Related Concepts

xbeta
Uniform convergence