Uniform convergence of Fourier Series satisfying Lipschitz condition

[rsa58]

Homework Statement

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.

Homework 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).
show:
\infty
\sum (|sin(nh)|^2) |a(n)|^2 \leq (K^2)*(h^2)
-\infty

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

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.

 

[rsa58]

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.)