Convergence of a Fourier Series with Convex Coefficients in $L^1(\Bbb T)$ (Katznelson's Ex $2.1.3.$)

Question

I want to show the following equivalence:

Suppose $\{a_n\}_{n\in \Bbb Z}$ satisfies $a_n \searrow 0$ as $|n|\to\infty$, $a_n = a_{-n} \ge 0$, $a_{n+1} + a_{n-1} - 2a_n \ge 0$ for $n > 0$. Then, the following are equivalent:

1. The partial sums $S_N(f)$ of the Fourier series of $f(t) = \sum_{n\in \Bbb Z} a_n e^{2\pi int}$ are bounded in $L^1(\Bbb T)$,
2. $a_n \log n = O(1)$ and the series $\sum_{n\in \Bbb Z} a_n e^{2\pi int}$ converges in $L^1(\Bbb T)$, and
3. $\lim_{n\to\infty}a_n\log n = 0.$

For $(3 \implies 2)$, it is clear that $a_n \log n \to 0$ implies $a_n \log n = O(1)$. As $a_n = a_{-n} \ge 0$ we must show that $\sum_{n\ge 2} \frac{\cos 2\pi nt}{\log n}$ converges in $L^1(\Bbb T)$ (I don't think this is true.) The implication $(2\implies 1)$ is immediate. As $S_N f \xrightarrow{L^1} f$, the partial sums $S_N f$ are bounded in the $L^1$ norm. $(1 \implies 3)$ seems to be the hardest. I believe we must use $\|D_N\|_1 \sim \log N$ where $D_N$ is the $N$th Dirichlet kernel.

Could I get some hints to complete $(3 \implies 2)$ and $(1 \implies 3)$? Thanks!

Answer 1

Certainly! You've already made some headway, so let's break down the implications and give some hints.

For ( (3 \implies 2) )

Given $a_n \log n \to 0$, you're right that $a_n \log n = O(1)$. This means that there exists some constant $ C $ such that $$|a_n \log n| \leq C$$ for sufficiently large $ n $. Now, to show that the series converges in $ L^1(\Bbb T) $, you might want to break down the summation into positive and negative indices and use properties of $ L^1 $ convergence.

Hint for ( (3 \implies 2) ):
Consider the absolute integrability of the terms, especially given that $ a_n = a_{-n} \ge 0 $ and the $O(1)$ behavior. Can you bound the integrals using the given conditions? Think about integrating term by term, and then switching the summation and integration (remember, this requires uniform convergence).

For ( (1 \implies 3) )

Given the boundedness of the partial sums $ S_N(f) $ in $ L^1(\Bbb T) $, and knowing that $$\|D_N\|_1 \sim \log N$$ for the Dirichlet kernel, you'll need to use properties of convolution and the fact that the Fourier coefficients of $ f $ are given by $ a_n $.

Hint for ( (1 \implies 3) ):
The Dirichlet kernel convoluted with $ f $ gives the partial sums of the Fourier series. Given that you know the $ L^1 $ norm behavior of both $ D_N $ and $ S_N(f) $, can you extract information about the Fourier coefficients $ a_n $, especially when multiplied by $ \log n $?

Additionally, make use of the properties of convolution in $ L^1 $, and remember that convolution with the Dirichlet kernel gives the partial Fourier series. The behavior of the Dirichlet kernel in $ L^1 $ and the given conditions might help you draw conclusions about the sequence $ \{a_n\} $.

Lastly, the problems involving Fourier series, especially questions about convergence, often require a good mix of analysis techniques and intuition. If you get stuck, it can be helpful to review the properties of Fourier coefficients, the behavior of the Dirichlet kernel, and properties of $ L^1 $ functions and their Fourier series.