Uniform convergence
In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set if, given any arbitrarily small positive number , an can be found such that each of the functions differ from by no more than at every point in . Loosely speaking, this means that converges to at a "uniform" speed on its entire domain, independent of .
Formulated more precisely, we say that converges to uniformly on if, given , there can be found an independent of , such that for all whenever . In contrast, we say that converges to pointwise, if there exists an , dependent on both and , such that when . It is clear from these definitions that uniform convergence of to on implies pointwise convergence for every .
The notation for uniform convergence of to is not quite standardized and different authors have used a variety of symbols including (in rough order of popularity) , , . Frequently, no special symbol is used, and authors simply write .
The difference between the two types of convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Weierstrass, is important because several properties of the functions , such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit 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 is independent of the variables and 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 is a set and is a real-valued function for . We say that the sequence is uniformly convergent with limit on if for every , there exists a natural number such that for all and all we have . Equivalently, converges uniformly on in the previous sense if and only if for every , there exists a natural number such that . This is the Cauchy criterion for uniform convergence.
In another equivalent formulation, if we define , then converges to uniformly if and only if as . This last statement can be restated as uniform convergence of on being equivalent to convergence of the sequence in the function space with respect to the uniform metric (also called the supremum metric), defined by .
The sequence is said to be locally uniformly convergent with limit if is a metric space and for every in , there exists an such that converges uniformly on . It is easy to see that local uniform convergence implies pointwise convergence. It is also clear that uniform convergence implies local uniform convergence.
Notes
Intuitively, a sequence of functions converges uniformly to if, given an arbitrarily small , we can find an so that the functions all fall within a "tube" of width centered around (i.e., between and ) for the entire domain of the function.
Note that interchanging the order of "there exists " and "for all " 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 converges pointwise with limit if and only if
- for every and every , there exists a natural number such that for all one has .
In explicit terms, in the case of uniform convergence, can only depend on , while in the case of pointwise convergence, may depend on both and . It is therefore plain that uniform convergence implies pointwise convergence. The converse is not true, as the example in the section below illustrates.
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 α_{0}, such that for every x in S and every α ≥ α_{0}: (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 converges to f uniformly if for all x in the domain of f* and all infinite n, is infinitely close to (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, with the uniform metric defined by . Then uniform convergence simply means convergence in the uniform norm topology: .
The sequence of functions with defined by is a classic example of a sequence of functions that converges to a function pointwise but not uniformly. To show this, we first observe that the pointwise limit of as is the function , given by
Pointwise convergence: Convergence is trivial for , since for all . For and given , we can ensure that whenever by choosing (here the upper square brackets indicate rounding up, see ceiling function). Hence, pointwise for all . Note that the choice of depends on the value of and . Moreover for a fixed choice of , (which cannot be defined to be smaller) grows without bound as approaches 1. These observations preclude the possibility of uniform convergence.
Non-uniformity of convergence: The convergence is not uniform, because given , no single choice of can ensure that for all , whenever . To see this, we note that regardless of how large becomes, there is always an such that (or any other positive value less than 1). Thus, if we choose , we can never find an such that for all and . Explicitly, given any candidate for , consider the value of at . Since , we have found an example of an that "escaped" our attempt to "confine" each to within of for all . In fact, it is easy to see that , contrary to the requirement that if .
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, , the limit 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 using the Weierstrass M-test.
Here is the series:
Any bounded subset is a subset of some disc of radius R, centered on the origin in the complex plane. The Weierstrass M-test requires us to find an upper bound on the terms of the series, with independent of the position in the disc:
To do this, we notice
and take .
If is convergent, then the M-test asserts that the original series is uniformly convergent.
The ratio test can be used here:
which means the series over is convergent. Thus the original series converges uniformly for all , and since , 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 is a compact interval (or in general a compact topological space), and is a monotone increasing sequence (meaning for all n and x) of continuous functions with a pointwise limit which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if is a compact interval and is an equicontinuous sequence that converges pointwise.
Applications
To continuity
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 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 is a sequence of functions, differentiable on , and such that converges for some point on . If converges uniformly on , then converges uniformly to a function , and for .
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 is a sequence of Riemann integrable functions defined on a compact interval I which uniformly converge with limit , then is Riemann integrable and its integral can be computed as the limit of the integrals of the :
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 is within ε of the graph of f, and so the upper sum and lower sum of are each within of the value of the upper and lower sums of , 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 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 converges for each x in E.
With this definition comes the following result: Theorem: Let x_{0} be contained in the set E and each f_{n} is continuous at x_{0}. If f = converges uniformly on E then f is continuous at x_{0} in E. Suppose that E = [a, b] and each f_{n} is integrable on [a, b]. If 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 converges almost uniformly on E if for every there exists a measurable set with measure less than such that the sequence of functions converges uniformly on . 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
Notes
- ↑ "Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem". Historia Mathematica. 32: 453–480. doi:10.1016/j.hm.2004.11.010.
- ↑ 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.
- ↑ Lakatos, Imre (1976). Proofs and Refutations. Cambridge University Press. p. 141. ISBN 0-521-21078-X.
- ↑ 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 , 3rd ed., Pearson, 2005
External links
- Hazewinkel, Michiel, ed. (2001), "Uniform convergence", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- "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