Pointwise and Uniform Convergence??

Question

Is there anyone who could please help me understand, our lecturers make things soooo complicated and I just want to understand what they mean in plain english instead of a load of stupid symbols!! Your help would be greatly appreciated thanks

Answer 1

This is a rather subtle issue but is crucial for quite a number of results. To get it, you need to know that definitions precisely:

Pointwise convergence: A sequence of functions f_n converges pointwise to a function f iff for every x in the domain and for every epsilon greater than 0, there is a natural number N so that n>=N implies that |f_n (x) -f(x)|
Uniform convergence: A sequence of functions f_n converges unifromly to a function f iff for every epsilon>0, there is a natrual number N so that for every x in the domain |f_n (x) -f(x)|
The difference is that the N in pointwise convergence may depend on the point x in the domain. Different x's may have different N's. For uniform convergence, a single N works for every point in the domain.

Now, if the doamin were a finite set, then knowing pointwise convergence would give a natural number N_k for each x_k in the domain. We could then define N=max{N_1, ...N_n} to get an N that works for all the domain. Hence, for a finite domain, pointwise and uniform convergence are the same.

If the domain is an infinite set, though, you can't simply take the maximum of all those N_k.

A good example to study: Let f_n (x)=x^n on the set [0,1). At each point x in the domain, the sequence x^n goes to 0. But, for any fixed n, there is an x value close to one where x^n is, say, 1/2. So no N will work for every x if we take epsilon=1/4. So this sequence of functions converges pointwise but not uniformly.

Answer 2

I will try to give you a "plain English" conceptual picture of this. But it will not be of help in solving problems -- for that you will need to go back to the rigorous definitions. I believe I saw it in an advanced calculus text somewhere.

Suppose we have a metal plate at room temperature, and we are going to heat it up. Let x be a variable describing points on the plate. Let Tt(x) be a sequence of functions describing the temperature of the plate at the time t, and at the point x.

We are told that the functions Tt converge, meaning that as t goes to infinity, the functions Tt approach a function T. This means the plate goes to a limiting temperature function over time. More rigorously, for any x, and for any chosen epsilon >0, |Tt (x)- T(x)| < epsilon if we choose t > N.

We heat the plate. Problem! Parts of the plate are converging to T(x) much faster (i.e., for lower t) than other parts, and the rate at which this is happening seems to depend on where you are on the plate. We are finding that the N we need to choose above depends upon x (where you are on the plate). Maybe this plate is not made of a uniform metal, and some parts heat faster than others do. This is non-uniform, or pointwise convergence.

We replace the plate with one made of the same metal throughout. Now we find that while different parts heat up at different rates, those rates do not depend on where you are on the plate. Our N does not depend upon x. The convergence is called uniform.

Answer 3

pointwise convergence: convergence at each individual point of a function (eventually lies within some band of width epsilon), but allows the strange behaviour that you have to go farther in the sequence to get some parts of the function within the epsilon band than others.

uniform convergence: the above strange behaviour doesn't happen. Given some epsilon band around the limit, eventually all parts of the function lie within that band.

There are many definitions for PC and UC, depending of what the sequence is, but they are all equivalent.

I must agree with the previous answerer: the formal definition is the best explanation. As hard as it may be, and as much as you may not want to, struggling over all the symbols now to try to understand exactly what they are saying will pay great dividends later.

Answer 4

For (a million, infinity), evaluate F1(x) = a million/x and Fk(x) = a million / kx, which exists for all n and converges to G(x) =0. seems to be uniformly convergent. For (0, infinity), evaluate F0(x) = a million / (0x) = does not exist. as a result, not uniformly convergent.

Answer 5

When I took my first course in analysis, I had the same gripe and fought tooth and nail against the mathematical definitions. The truth is that there is no better understanding than that which comes from the mathematical definition, because the definition is exact. Learn to use the symbols.

Answer 6

Sorry, I don't know! Ave...