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Converge pointwise with full Fourier series

  1. Feb 18, 2015 #1
    I am working on a simple PDE problem on full Fourier series like this:

    Given this piecewise function,

    ##f(x) =
    \begin{cases}
    e^x, &-1 \leq x \leq 0 \\
    mx + b, &0 \leq x \leq 1.\\
    \end{cases}##​

    Without computing any Fourier coefficients, find any values of ##m## and ##b##, if there is any, that will make ##f(x)## converge pointwise on ##-1 < x < 1## with its full Fourier series.

    I know for sure that if ##f(x)## is to converge pointwise with its full Fourier series, then ##f(x)## has to be piecewise smooth, meaning that each piece of ##f(x)## has to be differentiable.

    (a) Is this the right way to go?
    (b) If it is, how do you prove ##e^x## and ##mx + b## differentiable? By proving ##f'(x) = \lim_{x \to c}\frac{f(x) - f(c)}{x - c}## exists?

    Thank you for your time.
     
  2. jcsd
  3. Feb 18, 2015 #2

    Svein

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    1. f(x) has to be continuous at x = 0.
    2. f'(x) has to be continuous at x = 0.
     
  4. Feb 18, 2015 #3
    I think I am confused with the word "piecewise smooth." I had always thought it means "smooth piece by piece," meaning that ##f(x) = e^x## is smooth individually and then the next ##f(x) = mx +b## is smooth individually also. But your response implies that both parts of ##f(x)## have to be smooth as one big piece. So I am wrong on this? Let me know and thank you!
     
  5. Feb 18, 2015 #4

    Dick

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    No, you are right. That means there are no conditions on m and b. Notice there is a difference between saying "the series converges pointwise" and "the series converges pointwise to f(x)". If it's the latter you have a condition.
     
  6. Feb 18, 2015 #5
    What do you mean by "there are no conditions on ##m## and ##b##"? Thanks. [Nice to see you again! See, I had to tend one course after another! :-) ]
     
  7. Feb 18, 2015 #6

    Dick

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    I mean that it's piecewise smooth no matter what m and b are. Nice to see you!
     
  8. Feb 18, 2015 #7
    Thanks! I think it means ##m, b## are good for any real numbers. You are always omniscience from A to Z, omnipresent, and omni-helpful, if that is the right word.
     
  9. Feb 18, 2015 #8

    Ray Vickson

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    Your statement " .... ##f(x)## has to be piecewise smooth..." is false: it does not have to be piecewise smooth. It just has to obey the Dirichlet conditions; see, eg.,
    http://en.wikipedia.org/wiki/Dirichlet_conditions . These do not involve smoothness or differentiablility.

    So, with no restrictions on ##m,b## your function's Fourier series will converge pointwise on ##-1 \leq x \leq 1##, and will converge to ##f(x)## for ##-1 < x < 1, x \neq 0##. For some ##m,b## it will also converge to ##f(0)## when ##x = 0##, but for some other choices of ##m,b## it will converge to something else at ##x = 0## (but still converge).
     
  10. Feb 18, 2015 #9

    LCKurtz

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    @A.Magnus: I would almost bet that the original problem wants you to find m and b such that the FS converges pointwise to f(x). Otherwise there isn't much point to the problem. That would require specific values of m and b.
     
  11. Feb 18, 2015 #10
    I have uploaded the page that has the original problem 9, see the attached file. The text is "Introduction to Applied PDE" by John Davis, let me know if I got it very wrong in the first place, I will happily stand to be corrected. Also do let me know how should I go ahead if I was wrong. Thank you!

    PS:The text is extremely cut and dry, on top of that this is an online class, we get only reading assignments and homework, no lectures. Never complaining, so I take this site as crowd-teaching forum! :smile:
     

    Attached Files:

  12. Feb 18, 2015 #11

    Ray Vickson

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    The pdf displays upside-down on my screen, and I cannot rotate it (and so cannot read it). Anyway, have you read post #8?
     
  13. Feb 18, 2015 #12
    Yes, I did see #8, I am about to response. For the file, I will attached another one, give me just a second. Thanks, Ray!
     
  14. Feb 18, 2015 #13
    Ray, here is the corrected file. Feel free to J.Davis-PDE_Exercise9.png J.Davis-PDE_Exercise9.png crowd-teach me. Thanks.
     
  15. Feb 18, 2015 #14
    Ray, here is what I copy down verbatim from the John Davis' text, page 88:

    Let me know what I got wrong. Thanks again and again.
     
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