Uniform convergence

In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence { f_n } of functions converges uniformly to a limiting function f if the speed of convergence of f_n(x) to f(x) does not depend on x.

The concept is important because several properties of the functions f_n, such as continuity and Riemann integrability, are transferred to the limit f if the convergence is uniform.

History

Augustin Louis Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence.

The concept of uniform convergence was probably first used by Christoph Gudermann. Later his pupil Karl Weierstrass coined the term gleichmäßig konvergent (German: uniformly convergent) which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently a similar concept was used by Philipp Ludwig von Seidel and George Gabriel Stokes but without having any major impact on further development. G. H. Hardy compares the three definitions in his paper Sir George Stokes and the concept of uniform convergence and remarks: Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis.

Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others.

Definition

Suppose S is a set and f_n : S → R are real-valued functions for every natural number n. We say that the sequence (f_n) is uniformly convergent with limit f : S → R if for every ε > 0, there exists a natural number N such that for all x in S and all n ≥ N, |f_n(x) − f(x)| < ε.

Consider the sequence α_n = sup|f_n(x) − f(x)|. Clearly f_n goes to f uniformly if and only if α_n goes to 0.

The sequence (f_n) is said to be locally uniformly convergent with limit f if for every x in S, there exists an r > 0 such that (f_n) converges uniformly on B(x,r) ∩ S.

Notes

Compare uniform convergence to the concept of pointwise convergence: The sequence (f_n) converges pointwise with limit f : S → R if and only if

for every x in S and every ε > 0, there exists a natural number N such that for all n ≥ N, |f_n(x) − f(x)| < ε.

In the case of uniform convergence, N can only depend on ε, while in the case of pointwise convergence N may depend on ε and x. It is therefore plain that uniform convergence implies pointwise convergence. The converse is not true, as the following example shows: take S to be the unit interval [0,1] and define f_n(x) = xⁿ for every natural number n. Then (f_n) converges pointwise to the function f defined by f(x) = 0 if x < 1 and f(1) = 1. This convergence is not uniform: for instance for ε = 1/4, there exists no N as required by the definition.

Generalizations

One may straightforwardly extend the concept to functions S → M, where (M, d) is a metric space, by replacing |f_n(x) - f(x)| with d(f_n(x), f(x)).

The most general setting is the uniform convergence of nets of functions S → X, where X is a uniform space. We say that the net (f_α) converges uniformly with limit f : S → X iff

for every entourage V in X, there exists an α₀, such that for every x in I and every α≥α₀: (f_α(x), f(x)) is in V.

The above mentioned theorem, stating that the uniform limit of continuous functions is continuous, remains correct in these settings.

Examples

Given a topological space X, we can equip the space of bounded real or complex-valued functions over X with the uniform norm topology. Then uniform convergence simply means convergence in the uniform norm topology.

Properties

• Every uniformly convergent sequence is locally uniformly convergent
• Every locally uniformly convergent sequence is compactly convergent
• For locally compact spaces local uniform convergence and compact convergence coincide

Applications

to Continuity

Counterexample to a strengthening of the uniform convergence theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions $\scriptstyle \scriptstyle \sin ^{n}(x)$ converge to the non-continuous red function because convergence is not uniform

If $\scriptstyle S$ is a real interval (or indeed any topological space), we can talk about the continuity of the functions $\scriptstyle f_{n}$ and $\scriptstyle f$. The following is the more important result about uniform convergence:

Uniform convergence theorem. If $\scriptstyle (f_{n})_{n}$ is a sequence of continuous functions which converges uniformly towards the function $\scriptstyle f$, then $\scriptstyle f$ is continuous as well.

This is important, since pointwise convergence of continuous functions is not enough to guarantee continuity of the limit function as the image illustrates.

to Differentiability

If $\scriptstyle S$ is an interval and all the functions $\scriptstyle f_{n}$ are differentiable and converge to a limit $\scriptstyle f$, it is often desirable to differentiate the limit function $\scriptstyle f$ by taking the limit of the derivatives of $\scriptstyle f_{n}$. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable, and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. Consider for instance $\scriptstyle f_{n}(x)={\frac {1}{n}}\sin(nx)$ with uniform limit 0, but the derivatives do not approach 0. The precise statement covering this situation is as follows:

If $\scriptstyle f_{n}$ converges uniformly to $\scriptstyle f$, and if all the $\scriptstyle f_{n}$ are differentiable, and if the derivatives $\scriptstyle f'_{n}$ converge uniformly to g, then $\scriptstyle f$ is differentiable and its derivative is g.

to Integrability

Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, this can be done if uniform convergence is assumed:

If $\scriptstyle (f_{n})_{n=1}^{\infty }$ is a sequence of Riemann integrable functions which uniformly converge with limit $\scriptstyle f$, then $\scriptstyle f$ is Riemann integrable and its integral can be computed as the limit of the integrals of the $\scriptstyle f_{n}$.

Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the Lebesgue integral instead.

If $\scriptstyle S$ is a compact interval (or in general a compact topological space), and $\scriptstyle (f_{n})$ is a monotone increasing sequence (meaning $\scriptstyle f_{n}(x)\leq f_{n+1}(x)$ for all n and x) of continuous functions with a pointwise limit $\scriptstyle f$ which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if $\scriptstyle S$ is a compact interval and $\scriptstyle (f_{n})$ is an equicontinuous sequence that converges pointwise.

References

• Konrad Knopp, Theory and Application of Infinite Series; Blackie and Son, London, 1954, reprinted by Dover Publications, ISBN 0-486-66165-2.
• G.H. Hardy, Sir George Stokes and the concept of uniform convergence; Proceedings of the Cambridge Philosophical Society, 19, pp. 148-156 (1918)
• Bourbaki; Elements of Mathematics: General Topology. Chapters 5-10 (Paperback); ISBN 0-387-19374-X
• Walter Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1976.

External links

• "Uniform convergence". PlanetMath.
• "Limit point of function". PlanetMath.
• "Converges uniformly". PlanetMath.
• "Convergent series". PlanetMath.
• Graphic examples of uniform convergence of Fourier series from the University of Colorado