Convergence in Uniform and L2 sense, function interpretation

[trap101]

Let:


g_n(x) = 1 in [1/4 - 1/n² to 1/4 + 1/ n²) for n = odd
1 in [3/4-1/n² to 3/4 + 1/n²) for n = even
0 elsewhere

Show the function converges in the L2 sense but not pointwise.

My issue is in how I should use the definition of convergence because in all of the definitions of convergence between uniform, L2 and pointwise they all follow the similar rule of:

|f(x) - Ʃf_n(x)| --> 0

I am having issues in determining my f(x) for the difference because f_n(x) will be 1 depending on "n" being odd or even, so what would my exact value of f(x) be even though I am comparing the whole function?

 

[LCKurtz]

Let:


g_n(x) = 1 in [1/4 - 1/n² to 1/4 + 1/ n²) for n = odd
1 in [3/4-1/n² to 3/4 + 1/n²) for n = even
0 elsewhere

Show the function converges in the L2 sense but not pointwise.

You haven't told us on what interval. I'm guessing ##[0,1]##?

My issue is in how I should use the definition of convergence because in all of the definitions of convergence between uniform, L2 and pointwise they all follow the similar rule of:

|f(x) - Ʃf_n(x)| --> 0

I am having issues in determining my f(x) for the difference because f_n(x) will be 1 depending on "n" being odd or even, so what would my exact value of f(x) be even though I am comparing the whole function?

This problem has nothing to do with sums. Have you drawn a picture of the functions for n even and n odd? Does that give you an idea of what function ##g(x)## that ##g_n(x)## might converge to in ##L^2##? Can you calculate ##\|g_n-g\|_2## to get started?

 

[trap101]

You haven't told us on what interval. I'm guessing ##[0,1]##?



This problem has nothing to do with sums. Have you drawn a picture of the functions for n even and n odd? Does that give you an idea of what function ##g(x)## that ##g_n(x)## might converge to in ##L^2##? Can you calculate ##\|g_n-g\|_2## to get started?

Well first off I am going to assume that my "n" could only be positive integers. In the odd case the smallest value I could have for my "n" is 1 and in the even case it is 2, so my odd value interval startz at -3/4 and the even one starts at 1/2 since n goes to infiniti, these intervals end at 1/4 and 3/4 respectively. Now I just drew it, and on those intervals the function is a constant. So is it wise to interpret this as the function is 1 on these intervals, and the series is attempting to converge to this? In which case I could say f(x) = 1 and since the piece wise portions also equal 1 in terms of the series, using the defintion I could show the integrals all equal 0...thus convergence.

they did not give an interval so I am assuming -∞ to ∞

 

[LCKurtz]

Let me say it again: There is no summation and there is no series in this problem. You just have a sequence of functions and you are talking about whether ##g_n\to g## for some ##g(x)##, either pointwise or in ##L^2##. It doesn't matter whether the interval is ##[0,1]## or ##(-\infty,\infty)## because your functions are 0 except on short intervals.

Look at, for example, just even values of n. As those even values get larger, does ##g_n(x)## look like it converges pointwise to anything? Draw ##g_2,g_4,g_6## on the same graph and see what you think.

[Edit] I just noticed your title talks about uniform convergence but your post talks about pointwise vs. ##L^2##. They aren't the same thing, you know...

[Edit+] What about the sequences ##\{g_n(\frac 1 4)\}## and ##\{g_n(\frac 3 4)\}##?