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Littlewood's Tauberian theorem: Let $a_n=O(1/n)$. (In particular, given any $0<r<1$, the power series $\sum a_nr^n$ converges.) If the function defined by the power series $$f(r)=\sum a_nr^n$$ has a limit when $r$ tends to $1$ from below, then the power series also converges at $r=1$, and $f$ is continuous at $r=1$. That is, the series $\sum a_n$ is convergent, and its sum is $$\sum a_n = \lim_{r\nearrow 1} \sum a_nr^n\mbox{.}$$

I'm reading Stein & Shakarchi's Fourier Analysis. Given the above theorem, it is an exercise to prove the following:

(Problem 3iii, on chapter 2) If $f:\mathbb R\to\mathbb R$ is $2\pi$-periodic, continuous, and satisfies $\hat f(n)=O(1/n)$, then the Fourier series of $f$ converges uniformly to $f$.

Well, using:

  1. the fact that the Abel means of $f$ are given by the convolution of $f$ with the Poisson kernel $$P_r(x)=\sum_{n\in\mathbb Z}r^{|n|}e^{inx};$$
  2. the fact that the Poisson kernels form an approximation to the identity; and
  3. the continuity of $f$,

I can show that the Abel means $$P_r * f$$ converge uniformly to $f$ when $r\nearrow 1$. Therefore, the Abel means also converge pointwise, and then using Littlewood's theorem, I can show that the Fourier series of $f$ also converges pointwise. The problem is that I "lost" the information about the uniform convergence of the Abel means, since Littlewood's theorem is only concerned about the pointwise convergence of the series for each $x$, independently.

I know that the result in Problem 3iii, as stated above, does hold (that is, the exercise is not "mistaken" in any way). However, I can't see how this result could follow from Littlewood's theorem (I believe it must be a simple argument, but I can't see it). Any tips?

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    $\begingroup$ Could it be that the proof of Littlewood's theorem gives a bound on the tail of $\sum a_n$ that depends only on the rate of convergence of Abel sums and the constant in $O(1/n)$? If so, that would be enough. $\endgroup$ – user357151 Jul 16 '18 at 21:22
  • $\begingroup$ I'll investigated it, but I was ruling out this option, because the textbook doesn't provide the proof of the theorem. Instead, there's some notes hidden at the end of the book, referencing Titchmarsh's The Theory of Functions. I did read through the proof, but it's veeery lengthy. $\endgroup$ – fonini Jul 16 '18 at 22:17

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