Page 2 of 2 FirstFirst 12
Results 16 to 20 of 20

Math Help - Dense subset

  1. #16
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    Well my argument was was that the \theta i've got is a continuous linear functional, and \sin n is in the kernel for all n, except when n = 1. However \theta(\cos 2x) \not= 0 hence is not in the kernel. The kernel is closed by continuity, hence any sequence in it, in particular, a sequence of the sin functions, cannot converge to an element outside the closed set. The only problematic term is \sin which is not in the kernel, but that sequence is just \{\sin, \sin, \sin, ... \} which converges to \sin.

    I was just wondering where I have gone wrong here? Did you mean I should use Arzelą to find a fallacy in the above argument?
    Follow Math Help Forum on Facebook and Google+

  2. #17
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Rebesques View Post
    I'm not sure about this last argument Op, how can we restrict the interval to [0,pi]?
    The last part of the argument is that if \|s_n(g)-g\|_1 =  \int_{-\pi}^\pi |s_n(g)(x)-g(x)|\,dx \to0 then \|s_n(f)-f\|_1 =  \int_0^\pi |s_n(f)(x)-f(x)|\,dx \to0, because f(x)=g(x) when x\geqslant0. The integral of a positive function over part of an interval must be less than the integral over the whole interval.
    Follow Math Help Forum on Facebook and Google+

  3. #18
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    Let g be an odd function

    Then g(0) = -g(0) \implies g(0) = 0.

    Let f:[0, \pi] \rightarrow \mathbb{C}, given by f(x) = 1.

    Then f(0) \not= 0, hence cannot be extended to an odd function. So your method only works when f(0) = 0 and so it reasonable to believe the span is not dense

    But I don't know much of a difference this makes on the argument, because let's say we proceed with this constant function 1. then there is a discontinuity, does this stop us being able to talk about inner products, integrals etc?
    Follow Math Help Forum on Facebook and Google+

  4. #19
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by slevvio View Post
    Let g be an odd function

    Then g(0) = -g(0) \implies g(0) = 0.

    Let f:[0, \pi] \rightarrow \mathbb{C}, given by f(x) = 1.

    Then f(0) \not= 0, hence cannot be extended to an odd function. So your method only works when f(0) = 0 and so it reasonable to believe the span is not dense I claim that my method does work.

    But I don't know much of a difference this makes on the argument, because let's say we proceed with this constant function 1. then there is a discontinuity, does this stop us being able to talk about inner products, integrals etc? That's exactly correct. It doesn't matter if g has a discontinuity at 0. It still has a Fourier series, which converges to g in the L^2 and L^1 norms.
    Admittedly, I was being a bit sloppy in calling g an odd function. It becomes odd if we redefine g(0) to be 0. Changing the value a function at one point doesn't affect its Fourier series, and it doesn't change the L^1-norm.
    Follow Math Help Forum on Facebook and Google+

  5. #20
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    Opalg, I wasn't too sure about what you were saying since i only learned about the space of continuous functions, but I read some more and I see what you say is correct! So the span is dense. I spoke to my lecturer and we concluded that his original idea that the span wasn't dense is wrong.

    Here is why I was wrong. I claimed that the span could not contain any sequences that tend to cos2x since all the \sin n 's (except \sin 1) were in a closed subset and cos2x isnt. However throwing sin in gives a sequence which does converge i.e. the partial sums of the fourier series

    Thanks everyone for this very long and informative thread
    Follow Math Help Forum on Facebook and Google+

Page 2 of 2 FirstFirst 12

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: August 28th 2011, 02:33 AM
  2. Dense subset
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 1st 2010, 08:47 AM
  3. Dense subset
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: February 8th 2010, 06:51 PM
  4. Dense subset of R^n
    Posted in the Math Challenge Problems Forum
    Replies: 9
    Last Post: January 28th 2010, 09:12 AM
  5. Dense subset
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 3rd 2008, 10:38 AM

Search Tags


/mathhelpforum @mathhelpforum