Uniform convergence is a type of convergence of a sequence of real valued functions requiring that the difference to the limit function can be estimated uniformly on , that is, independently of . This condition makes uniform convergence a stronger type of convergence than pointwise convergence.
Given a convergent sequence of functions , it is natural to examine the properties of the resulting limit function . It turns out that the uniform convergence property implies that the limit function inherits some of the basic properties of , such as continuity, boundedness and Riemann integrability, in contrast to some examples of the limit function of pointwise convergence.
The functions , converge pointwise to the function All of the functions are continuous but is discontinuous. The convergence is not uniform.
Uniform convergence simplifies certain calculations, for instance by interchanging the integral and the limit sign in integration.
Difficulties which arise when the convergence is pointwise but not uniform can be seen in the example of the non Riemann integrable indicator function of rational numbers in and provide partial explanations of some other anomalies such as the Gibbs phenomenon. Many theorems of functional analysis use uniform convergence in their formulation, such as the Weierstrass approximation theorem and some results of Fourier analysis. Uniform convergence can be used to construct a nowhere-differentiable continuous function.
A sequence of real-valued functions on a set , , is said to be uniformly convergent on to a limit function if for each there exists an such that
holds for all and .
The crucial condition which distinguishes uniform convergence from pointwise convergence of a sequence of functions is that the number in the definition depends only on and not on . It follows that every uniformly convergent sequence of functions is pointwise convergent to the same limit function, thus uniform convergence is stronger than pointwise convergence.
The definition of the uniform convergence is equivalent to the requirement that
If we denote by the supremum norm of a function , the last condition becomes
Find a sequence of functions which converges pointwise but not uniformly.
Let be a sequence of functions . Then
so the sequence converges pointwise to the function
However, given an , for we have so there can not exist an for which the estimate will hold uniformly (that is, for all ) and the convergence is not uniform. The last argument can be replaced by observing that for all (for arbitrary close to we have that is arbitrary close to 1). This provides a quicker way to conclude that the convergence is not uniform, since .
The following theorem shows that the uniform convergence behaves well under addition and multiplication:
Let be two sequences of functions , respectively, which converge uniformly to functions and on . Then:
the sequence converges uniformly to on
if then the sequence converges uniformly to on
The notion of uniform convergence extends to series of functions:
A series , with is said to converge uniformly to a function on if the sequence of partial sums, converges uniformly to .
Fortunately, there exists a criterion for the uniform convergence of series which is easy to apply in practice:
Let be a sequence of real valued functions on a set such that there is a sequence of positive numbers satisfying If , then the series converges uniformly on .
Riemann Zeta function
For let denote the Riemann Zeta function on . Fix and consider the Zeta function on as a series of functions on . For each , implies and the series converges so the Weierstrass M-test implies that is uniformly convergent on .
When and are equipped with their natural topologies, the sequence of functions consists of continuous functions while their pointwise limit is discontinuous. However, if the convergence is uniform such phenomena do not appear:
Uniform limit theorem
Let be a topological space and equipped with the natural topology. If a sequence of continuous functions converges uniformly to a function , then is continuous.
Restricting the sequence of functions on the set produces the continuous pointwise (but not uniform) limit on , which shows that the continuity of the limit function is not a sufficient condition to imply the uniform convergence of a sequence of continuous functions. There is, however, a partial result in this direction, Dini's theorem which, under additional assumptions that the underlying space is compact and that the sequence is monotone, shows that the continuity of the limit function implies uniform convergence.
Riemann Zeta function is continuous on
is the uniformly convergent limit of its partial sums on each , , which are continuous functions. Hence, by the Uniform limit theorem is continuous at each .
Consider the functions defined by . For every there is a natural number such that for each (take ). It follows that the pointwise limit of is the function given by . Functions are all bounded functions () but the limit function is unbounded. In the case of uniform convergence of bounded functions, the limit function is again bounded:
Uniform convergence of bounded functions
If a sequence of bounded functions converges uniformly to a function , then is bounded.
For the series converges pointwise to the function . The partial sums are bounded functions while the limit function is unbounded on . Hence the geometric series does not converge uniformly on .
It can however be shown that the geometric series converges uniformly on each interval for .
As a countable set, can be enumerated by . Let be the indicator function of the set , that is Each is Riemann integrable as a bounded function with finitely many discontinuities on . However, the pointwise limit of is the indicator function of , which is known to be non Riemann integrable.
In the example above the pointwise limit of Riemann integrable functions turned out to be non Riemann integrable. With the assumption of uniform convergence the following theorem holds:
Uniform convergence and Riemann integrability
Let be a sequence of Riemann integrable functions on converging uniformly to . Then is Riemann integrable and
In particular, if the series converges uniformly, then
An expansion for the logarithmic function
As stated above, the geometric series is uniformly convergent in each interval , . Applying this to the Riemann integral we can interchange the summation and the integral in the following calculation:
Weierstrass Function is continuous
In 1872, Karl Weierstrass published a paper on a function that is continuous and nowhere differentiable, thus giving a new counter-intuitive insight into the relation of continuity and differentiability. He analyzed the function
and proved, under the assumptions that , is an odd integer and , that the function is continuous and nowhere differentiable. While the proof of non differentiability is beyond the scope of this text, continuity follows directly from the Uniform limit theorem:
For all we have and for so, by the Weierstrass M-Test the defining series for is uniformly convergent. Functions are continuous and so are the partial sums of the series for . By the Uniform limit theorem, is continuous.
Note: this example also shows that the uniform limit of differentiable functions need not be differentiable.