Pointwise and Uniform Convergence

In summary: Yes, I understand the concept of uniform convergence for a sequence of real valued functions of real variable.
  • #1
Silviu
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Hello! Can someone explain to me in an intuitive way the difference between pointwise and uniform convergence of a series of complex functions ##f_n(z)##? Form what I understand, the difference is that when choosing an "N" such that for all ##n \ge N## something is less than ##\epsilon##, in the case of uniform convergence N can't depend on z but in the case of pointwise convergence it can. I saw an example for the function ##f_n(z)=z^n## which has uniform convergence if the domain is ##D_{[0,a]}## with ##a<1##, but has pointwise convergence for ##a=1##. I understand that the proofs involves different approaches and this is why they have different types of convergence, but in the end they converge to the same function ##f(z)=0##. So how is the uniform convergence stronger than the other one (I know that pointwise is implied by uniform but not the other way around, but this still doesn't really give me a clear understanding). Thank you!
 
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  • #2
If you have a function ##f_n (x) = x^n## on the half-open interval ##[0,1[##, the sequence ##f_1 (a),f_2 (a), f_3 (a), \dots## will approach zero no matter what the number ##a\in [0,1[## is, but for any given ##n##, no matter how large, there is some number ##x\in [0,1[## for which ##f_n (x) = 0.5## or any number in the interval ##[0,1[##. So you can't limit the value of ##f_n (x)## to some interval ##[0,c]## where ##c< 1## by any finite choice of ##n##.
 
  • #3
The main thing about uniform convergence that I know is that the uniform limit of continuous functions is continuous. The point wise limit may not be continuous.
 
  • #4
Silviu said:
Hello! Can someone explain to me in an intuitive way the difference between pointwise and uniform convergence of a complex function ##f(z)##? Form what I understand, the difference is that when choosing an "N" such that for all ##n \ge N## something is less than ##\epsilon##,

Are you talking about uniform convergence "of a complex function"? - or uniform convergence of a sequence of functions? It would be best to ask a precise question, even if you want an intuitive explanation.
 
  • #5
Stephen Tashi said:
Are you talking about uniform convergence "of a complex function"? - or uniform convergence of a sequence of functions? It would be best to ask a precise question, even if you want an intuitive explanation.
Sorry, I meant series of functions (anyway for a single function I guess it doesn't make a difference as there is no "n" to take the limit of, just z)
 
  • #6
Silviu said:
Sorry, I meant series of functions
I think you mean "sequence" of functions.

(anyway for a single function I guess it doesn't make a difference as there is no "n" to take the limit of, just z)

I don't what "it" refers to.

Do you understand the concept of uniform convergence for a sequence of real valued functions of real variable?
 

Related to Pointwise and Uniform Convergence

What is the difference between pointwise and uniform convergence?

Pointwise convergence refers to the convergence of a sequence of functions at each individual point in the domain. Uniform convergence, on the other hand, refers to the convergence of the sequence of functions as a whole, meaning that the functions get closer together uniformly across the entire domain.

How do you determine if a sequence of functions converges pointwise or uniformly?

To determine if a sequence of functions converges pointwise or uniformly, you can use the definition of each type of convergence. For pointwise convergence, you need to show that for every point in the domain, the sequence of function values approaches the limit as the index of the sequence goes to infinity. For uniform convergence, you need to show that for any given small number, there exists an index in the sequence after which all the function values are within that small number of the limit.

Why is uniform convergence considered stronger than pointwise convergence?

Uniform convergence is considered stronger than pointwise convergence because it guarantees that the functions in the sequence get closer to each other at a uniform rate across the entire domain. This means that the limit function is not only approached at every point, but also that the rate of convergence is the same at every point.

What are some examples of functions that converge pointwise but not uniformly?

One example of a sequence of functions that converges pointwise but not uniformly is the sequence of functions f_n(x) = x^n on the interval [0,1]. At every point in the interval, the function values approach 0 as n goes to infinity, but the rate of convergence varies at different points, making it not uniformly convergent. Another example is the sequence of functions f_n(x) = 1/n on the interval [0,1]. Again, the function values converge to 0 at every point, but the rate of convergence is not the same at every point.

How can uniform convergence be useful in applications?

Uniform convergence can be useful in applications because it guarantees that the limit of the sequence of functions is also continuous. This makes it easier to work with in applications where continuity is important, such as in numerical analysis and optimization problems. Additionally, uniform convergence allows for the interchange of limits and integrals, making it a useful tool in solving certain types of problems.

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