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, but not necessarily if the convergence is not.

Uniform convergence to a function on a given interval can be defined in terms of the uniform norm.

History

In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions.^[1]

The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series $\textstyle{\sum_{n=1}^\infty f_n(x,\phi,\psi)}$ is independent of the variables $\phi$ and $\psi.$ While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.^[2]

Later Gudermann's pupil Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, 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^[3] 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 $f_n: S \to \mathbb R$ is a real-valued function for every natural number $n$. We say that the sequence $(f_n)_{n \in \mathbb N}$ is uniformly convergent with limit $f: S \to \mathbb R$ if for every $\varepsilon > 0$, there exists a natural number $N$ such that for all $x \in S$ and all $n \geq N$ we have $|f_n(x) - f(x)| < \varepsilon$.

Consider the sequence $a_n = \sup_x |f_n(x) - f(x) |$ where the supremum is taken over all $x \in S$. Then $f_n$ converges to $f$ uniformly if and only if $a_n$ tends to 0.

The sequence $(f_n)_{n \in \mathbb N}$ is said to be locally uniformly convergent with limit $f$ if $S$ is a metric space and for every $x$ in $S$, there exists an $r > 0$ such that $(f_n)$ converges uniformly on $B(x,r) \cap S$.

Notes

Note that interchanging the order of "there exists N" and "for all x" in the definition above results in a statement equivalent to the pointwise convergence of the sequence. That notion can be defined as follows: the sequence (f_n) converges pointwise with limit f : S → R if and only if

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

Here the order of the universal quantifiers for x and for ε is not important, but the order of the former and the existential quantifier for N is.

In the case of uniform convergence, N can only depend on ε, while in the case of pointwise convergence N may depend on both 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. This is because solving for n gives n > log ε / log x. This depends on x as well as on ε. Also note that it is impossible to find a suitable bound for n that does not depend on x because for any nonzero value of ε, log ε / log x grows without bounds as x tends to 1.

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 if and only if

for every entourage V in X, there exists an α₀, such that for every x in S 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.

Definition in a hyperreal setting

Uniform convergence admits a simplified definition in a hyperreal setting. Thus, a sequence $f_n$ converges to f uniformly if for all x in the domain of f* and all infinite n, $f_n^*(x)$ is infinitely close to $f^*(x)$ (see microcontinuity for a similar definition of uniform continuity).

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.

The sequence $f_n:[0,1]\rightarrow [0,1]$ with $f_n(x):=x^n$ converges pointwise but not uniformly:

$\lim_{n\rightarrow \infty} f_n(x) = \begin{cases} 0, & x \in [0,1) \\ 1, & x=1. \end{cases}$

In this example one can easily see that pointwise convergence does not preserve differentiability or continuity. While each function of the sequence is smooth, that is to say that for all n, $f_n\in C^{\infty}([0,1])$, the limit $\lim_{n\rightarrow \infty}f_n$ is not even continuous.

Exponential function

The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset S of $\mathbb{C}$ using the Weierstrass M-test.

Here is the series:

$\sum_{n=0}^{\infty}\frac{z^n}{n!}.$

Any bounded subset is a subset of some disc $D_R$ of radius R, centered on the origin in the complex plane. The Weierstrass M-test requires us to find an upper bound $M_n$ on the terms of the series, with $M_n$ independent of the position in the disc:

$\left| \frac{z^n}{n!}\right|\le M_n , \forall z\in D_R.$

This is trivial:

$\left| \frac{z^n}{n!}\right| \le \frac{\left| z\right|^n}{n!} \le \frac{R^n}{n!}$

$\Rightarrow M_n=\frac{R^n}{n!}.$

If $\sum_{n=0}^{\infty}M_n$ is convergent, then the M-test asserts that the original series is uniformly convergent.

The ratio test can be used here:

$\lim_{n \to \infty}\frac{M_{n+1}}{M_n}=\lim_{n \to \infty}\frac{R^{n+1}}{R^n}\frac{n!}{(n+1)!}=\lim_{n \to \infty}\frac{R}{n+1}=0$

which means the series over $M_n$ is convergent. Thus the original series converges uniformly for all $z\in D_R$, and since $S\subset D_R$, the series is also uniformly convergent on S.

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.
• A sequence of continuous functions on metric spaces, with the image metric space being complete, is uniformly convergent if and only if it is uniformly Cauchy.
• If $S$ is a compact interval (or in general a compact topological space), and $(f_n)$ is a monotone increasing sequence (meaning $f_n(x) \leq f_{n+1}(x)$ for all n and x) of continuous functions with a pointwise limit $f$ which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if $S$ is a compact interval and $(f_n)$ is an equicontinuous sequence that converges pointwise.

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 $\sin^n(x)$ converge to the non-continuous red function. This can happen only if convergence is not uniform.

If I is a real interval (or indeed any topological space), we can talk about the continuity of the functions f_n and f. The following is the more important result about uniform convergence:

Uniform convergence theorem. If (f_n) is a sequence of continuous functions all of which are defined on the interval I which converges uniformly towards the function f on an interval I, then f is continuous on I as well.

This theorem is proved by the "ε/3 trick", and is the archetypal example of this trick: to prove a given inequality (ε), one uses the definitions of continuity and uniform convergence to produce 3 inequalities (ε/3), and then combines them via the triangle inequality to produce the desired inequality.

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

More precisely, this theorem states that the uniform limit of uniformly continuous functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous.

To differentiability

If S is an interval and all the functions f_n are differentiable and converge to a limit f, it is often desirable to differentiate the limit function f by taking the limit of the derivatives of f_n. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable (not even if the sequence consists of everywhere-analytic functions, see Weierstrass function), 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 $f_n(x) = \frac1n \sin(nx)$ with uniform limit 0, but the derivatives do not approach 0. In order to ensure a connection between the limit of a sequence of differentiable functions and the limit of the sequence of derivatives, the uniform convergence of the sequence of derivatives plus the convergence of the sequence of functions at at least one point is required. The precise statement covering this situation is as follows:^[4]

Suppose $\scriptstyle {f_n}$ is a sequence of functions, differentiable on $\scriptstyle [a, b]$, and such that $\scriptstyle {f_n(x_0)}$ converges for some point $\scriptstyle x_0$ on $\scriptstyle [a, b]$. If $\scriptstyle f'_n$ converges uniformly on $\scriptstyle [a, b]$, then $\scriptstyle {f_n}$ converges uniformly to a function $\scriptstyle f$, and $\scriptstyle f'(x) = \lim_{n\to \infty} f'_n(x)$ for $\scriptstyle x \in [a, b]$.

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 $(f_n)_{n=1}^\infty$ is a sequence of Riemann integrable functions defined on a compact interval I which uniformly converge with limit $f$, then $f$ is Riemann integrable and its integral can be computed as the limit of the integrals of the $f_n$:

$\int_I f = \lim_{n\to\infty}\int_I f_n.$

In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function. This follows because, for n sufficiently large, the graph of $f_n$ is within ε of the graph of f, and so the upper sum and lower sum of $f_n$ are each within $\varepsilon |I|$ of the value of the upper and lower sums of $f$, respectively.

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.

To analyticity

If a sequence of analytic functions converges uniformly in a region S of the complex plane, then the limit is analytic in S. This demonstrates an example that complex functions are more well-behaved than real functions, since the uniform limit of analytic functions on a real interval need not even be differentiable (see Weierstrass function).

To series

We say that $\textstyle\sum_{n=1}^\infty f_n$ converges:

i) pointwise on E if and only if the sequence s_n converges where s_n(x) is the sequence of partial sums.

ii) uniformly on E if and only if s_n(x) converges uniformly as n goes to infinity.

iii) absolutely on E if and only if $\textstyle\sum_{n=1}^\infty |f_n|$ converges for each x in E.

With this definition comes the following result: Theorem: Let x₀ be contained in the set E and for each f_n is continuous at x₀. If f = $\textstyle\sum_{n=1}^\infty f_n$ converges uniformly on E then f is continuous at x₀ in E. Suppose that E = [a, b] and each f_n is integrable on [a, b]. If $\textstyle\sum_{n=1}^\infty f_n$ converges uniformly on [a, b] then f is integrable on [a, b] and the series of integrals of f_n is equal to integral of the series of f_n. This is known as term by term integration.

Almost uniform convergence

If the domain of the functions is a measure space E then the related notion of almost uniform convergence can be defined. We say a sequence of functions $(f_n)$ converges almost uniformly on E if for every $\delta > 0$ there exists a measurable set $E_\delta$ with measure less than $\delta$ such that the sequence of functions $(f_n)$ converges uniformly on $E \setminus E_\delta$. In other words, almost uniform convergence means there are sets of arbitrarily small measure for which the sequence of functions converges uniformly on their complement.

Note that almost uniform convergence of a sequence does not mean that the sequence converges uniformly almost everywhere as might be inferred from the name.

Egorov's theorem guarantees that on a finite measure space, a sequence of functions that converges almost everywhere also converges almost uniformly on the same set.

Almost uniform convergence implies almost everywhere convergence and convergence in measure.

See also

• Modes of convergence (annotated index)
• Dini's theorem

Notes

1. "Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem". Historia Mathematica 32: 453–480. doi:10.1016/j.hm.2004.11.010.
2. Jahnke, Hans Niels (2003). "6.7 The Foundation of Analysis in the 19th Century: Weierstrass". A history of analysis. AMS Bookstore. ISBN 978-0-8218-2623-2, p. 184.
3. Lakatos, Imre (1976). Proofs and Refutations. Cambridge University Press. p. 141. ISBN 0-521-21078-X.
4. Rudin, Walter. Principles of Mathematical Analysis Third edition. 1976. McGraw-Hill International editions.

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.
• Gerald Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999, ISBN 0-471-31716-0.
• William Wade, An Introduction to Analysis <\cite>, 3rd ed., Pearson, 2005

External links

• Hazewinkel, Michiel, ed. (2001), "Uniform convergence", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Uniform convergence at PlanetMath.org.
• Limit point of function at PlanetMath.org.
• Converges uniformly at PlanetMath.org.
• Convergent series at PlanetMath.org.
• Graphic examples of uniform convergence of Fourier series from the University of Colorado