Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bounded integrable periodic function

  1. Jan 16, 2010 #1
    hi,
    i have a hard problem, i guess so,
    i am looking for any help
    Code (Text):

    g(x) is a bounded Lebesgue measurable function that is periodic
    i.e.  [tex]g(x)=g(x+p)[/tex]. Then for every [tex]f \in L^1(\Re)[/tex]

    [tex]lim_{n\rightarrow \infty}\int_{\Re}f(x)g(nx) dx=(\int_{\Re}f(x)dx)((1/p){\int_{0}^{p}g(x) dx)[/tex]
    thanks for any help
     
  2. jcsd
  3. Jan 16, 2010 #2

    mathman

    User Avatar
    Science Advisor

    You can use the fact that g(nx+p)=g(n(x+p/n)). Therefore the integral over shorter and shorter intervals will contain a full cycle of g leading to the result described. I'll let you work out the details.
     
  4. Jan 16, 2010 #3
    after 4 hours working :)
    i solved , here the steps
    1-) change of u=nx and as n goes to infty [-n,n] goes to whole space
    2-) after some operations and by dividing [-n,n] into 2n/p equal pieces with lenght p/n
    3-) then again change of u=t+ip-p where -n^2/p < i < n^2/p,
    4-) by riemann integration as n goes to infinity, we get right hand side

    that was real fun to solve it
    if i cant solve , then it becomes a huge pain :)
     
  5. Jan 17, 2010 #4

    mathman

    User Avatar
    Science Advisor

    The main quibble I have with your approach is that the original problem was in terms of f being Lebesgue integrable, not Riemann.
     
  6. Jan 18, 2010 #5
    the set of continuous functions is dense in the set of Lebesgue integrable functions

    So when we do the problem for a continuous function (i.e. riemann integrable)
    we can extend it to any lebesgue integrable function.

    am i wrong?
     
  7. Jan 18, 2010 #6

    mathman

    User Avatar
    Science Advisor

    I'm pretty rusty on the subject, but you are probably right.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook