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Uniform convergence of Fourier Series satisfying Lipschitz condition

  1. Nov 22, 2008 #1
    1. The problem statement, all variables and given/known data

    f is integrable on the circle and satisfies the Lipschitz condition (Holder condition with a=1). Show that the series converges absolutely (and thus uniformly). i literally spent about 20 hours on this problem today but i just could not figure it out. i have a feeling it's not that hard, but i am having a hell of a time estimating the sum that i need to show the fourier coefficients converge absolutely.

    2. Relevant equations
    |f(x) - f(y)| <= K |x-y| for all x, y.
    I have done up to and including this part of the problem (the easiest part).
    [tex]\sum[/tex] (|sin(nh)|^2) |a(n)|^2 [tex]\leq[/tex] (K^2)*(h^2)

    now take h = pi/(2^p+1) and show: (i'm going to call this inequality ***)
    [tex]\sum[/tex] |a(n)|^2 [tex]\leq[/tex] (K^2)(pi^2)/(2^2p+1)
    (2^p-1) < |n| <= 2^(p)

    where a(n) is the nth Fourier coefficient. No need to comment on the above.

    3. The attempt at a solution
    Here's the part i can't get: estimate the partial sums of the Fourier coefficients where the absolute value of the summation index n ranges over 2^(2p-1) to 2^p as above, in order to show that the Fourier series converges absolutely. Hint: Use the Cauchy Schwartz inequality.

    I applied the Cauchy Schwartz inequality to the inner product of f and the complex exponential to get a bound for this quantitiy which is the L^2 norm of f. this is ofcourse the inner product of f with itself square root. but the inner product of the partial sums of the fourier series must converge to the inner product of f with itself, and this inner product is akin to the left side of *** so that the fourier coefficients must converge to zero. anyway, this is not the answer that i need because i need to estimate the partial sums of the fourier coefficients in the given range. i tried lots of other stuff too, like moving various sums inside the inner product, but i feel there is something fundamental that i am missing. please give me hints or references but not solutions.

    PS: i also tried summing *** over 1 to N so that the left side of *** becomes the partial sums of the inner product of the two sided sequence of fourier coefficients in the little L over z (sorry don't really know how to say this) norm with itself [i mean the vector space of all two sided sequences whose infinite sum of absolute values of components converges]; the right side becomes a geometric series ofcourse. blah, sorry i am kind of lost in this course.
  2. jcsd
  3. Nov 23, 2008 #2
    hey everybody i solved it so don't worry, i will be posting the solution in a couple days here for those who are interested. i was allowing p to vary instead of keeping it fixed. The one thing i wasn't sure about was interchanging a finite sum with the infinite sum of an absolutely convergent series. however i think it's okay because with absolute convergence we can sum in which ever order we want... (i am talking about the inner product in the little L squared Z vector space.)
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