Dirichlet conditions for the Fourier series

Discussion in 'Math' started by jagjit Sehra, Mar 20, 2008.

  1. jagjit Sehra

    jagjit Sehra Thread Starter Member

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    Hi Folks,
    I have read about the three conditions that must be satisfied before the Fourier series can be successfully used to represent any given function.
    Can someone please confirm that my understanding of them is correct!

    1)

    The function must have a single defined value for each value in its
    domain.

    For every value of x there should be only one value of y. Most of the electronic (This is my field) functions will satisfy this condition. One function that wouldn't satisfy this condition is the circle function (sometimes not reffered as a function because two values are obtained for x) because that would give two values for each value of x.

    2)
    The function must not have any infinite discontinuity of range within its domain.

    This condition states that the function shouldn't have an infinite number of discontinuities on its domain. It's kind of hard to come up with a function that doesn't abide to this condition. One example would be:

    let f be a function of x so that its value is:
    1 if x is a rational number
    0 if x is an irrational number

    This function has an infinite number of discontinuities, because in any
    finite interval, there is an infinite number of rational numbers, and an infinite number of irrational numbers between them! So the function would have an infinite number of "jumps" between 0 and 1.
    I guess if a function did have a lot of jump the Fourier representaion would be skewed by the Gibbs Phenomina.

    The first and second differentials of the function must be piecewise
    continuous over the domain.

    a. within each interval, the function is continuous; and
    b. on the "borders" of each interval, the function can be discontinuous, as long as the limit coming from each side exists. For example, the tan(x) function is NOT piecewise continuous, because near pi/2 multiples it
    "Points towards" infinity. Had it pointed towards any real value, it would be piecewise continuous.


    Thank you in advance for you guidance and help.

    Jag.
  2. studiot

    studiot E-book Developer

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    My, you have been doing your homework dude.

    But that's only two conditions, by my counting.

    1)
    Exactly so.
    The standard parabola y^2=4ax also gets caught here.

    2.) Condition correct but you only need consider the interval of interest not the whole domain of the function.

    3)

    The third condition is that there must be only a finite number of maxima and minima on the function in the interval concerned.

    A function that violates this is

    y=sin(1/x) 0<x<=1

    y=0 -1<=x<=0

    You may be interested to know that the fourier series at any point converges to the average of the convergenge approaching from the left and from the right. If these are equal the convergence is exact,
  3. Mark44

    Mark44 Well-Known Member

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    If it's referred to as a function, that is an error. By definition, a function in the usual sense cannot have more than one value. By "usual sense" I am excluding functions whose domain and/or range is multidimensional.

    When you say "infinite discontinuity of range" it's not clear to me what you mean. The function f(x) = 1/x has an infinite discontinuity at x = 0, but x = 0 is not in the domain of this function. The example you give has an infinite number of discontinuities, as you point out, but doesn't fall into the "infinite discontinuity of range", unless you are defining this in some way that I don't understand.

    I found another definition of Dirichlet Conditions at http://mathworld.wolfram.com/DirichletFourierSeriesConditions.html, a site associated with Wolfram, the creator of Mathematica.

    Mark
  4. Mark44

    Mark44 Well-Known Member

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    Right, since this is not a function. However, you can break it up into two pieces, each of which is a function, and each of which satisfies all the conditions.

    y1 = +sqrt(4*a*x)
    y2 = -sqrt(4*a*x)

    Mark
  5. studiot

    studiot E-book Developer

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    I think the link definition implies jagjit's condition1

    What's a 'regular' function?

    The Dirichlet conditions spring from a wider Dirichlet theorem about the convergence of series.
  6. Mark44

    Mark44 Well-Known Member

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    There's a hyperlink on this term that goes to a definition.
  7. studiot

    studiot E-book Developer

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    Thanks

    This is unnecessarily restrictive, but certainly incorporates the single valued bit. That is
    jagjits condition 1 as I suspected.

    It also implies there are functions that are not single valued or there would be no need to state this.

    It should be remembered that when Mr Dirichlet was around multivalued functions were admitted to the fraternity of functions. So this condition made sense.

    It has become fashionable to disallow multivalued functions in certain branches of mathematics, although they are certainly still acknowledged in complex analysis.

    It is really a matter of definition.

    The important thing, for which Jagjit should be congratulated, is that (s)he is looking at the conditions under which a theorem applies. These are all too often glossed over on the basis that 'you will never meet an exception in the exam' . So when one crops up in real life, confusion reigns.
  8. jagjit Sehra

    jagjit Sehra Thread Starter Member

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    Hi Folks,
    Firstly I would like to thank all the good people who helped me on my last post (Dirchlet conditions).
    I recently heard where they played a recording of someone singing she didnt sound very good, however after the studio had done something to it the same song didnt sound too bad. It sounded a lot more crisp.
    I have only started out in this field, but has this got something to do with Fourier. I understand that it cant be the Fourier series as the signal must be periodic. It could be the Fourier transform though, could they de-compose the audio signal segment into its composite sine and cosine waves and just reconstruct the signal using the inverse transform. When reconstructing could they not put back certain frequencies (certain notes that couldnt be reached).
    If that is possible then could they also put frequencies in that wernt there at all. I cant see how else they would do it. What I am saying is very abstract, I have heard that this kind of stuff falls into the DSP domain but is my thinking right.

    P.S.
    Happy Easter to everyone.

    Cheers Jag.
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