I am hoping to prove a version of Lemma 1.12 about convex Fourier coefficients from Classical and Multilinear Harmonic Analysis by Muscalu and Schlag. The lemma is as follows:
Let $\{a_n\}_{n \in \mathbb{Z}}$ be an even sequence of nonnegative numbers that tend to zero, which is convex in the following sense: $$a_{n+1} + a_{n−1} − 2a_n \geq\ 0 \forall\ n > 0.$$ Then, there exists $f \in L^1(\mathbb{T})$ with $f \geq 0$ and $\hat{f}(n) = a_n$.
Here $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. The crux of the proof is in noticing that the Fourier coefficients of the Fejer kernels are exactly $\hat{K}_n(m) = (n-m)$. This allows them to construct the function
$$f = \sum_{n=1}^\infty n(a_{n+1} + a_{n-1} + 2a_n)K_n,$$
which has a bunch of nice properties which prove the lemma, and very important has bounded norm, $\Vert f \Vert_{L^1(\mathbb{T})} = \vert \hat{f}(0) \vert = a_0 < \infty$.
The lemma can be modified to work for odd convex sequences if one takes a family of kernels $F_n$ such that $\hat{F}_n(m) = \hat{K}_n(m)$ for $m \geq 0$ and $\hat{F}_n(m) = -\hat{K}_n(m)$ for $m < 0$. The issue I am having in proving this is that I don't know what the $L^1$ norm of these kernels are, in particular, whether they are bounded. If they are then that would be great and the proof is done by the exact same argument as in M&S. If not, then it may be because the claim is false... (This is in hopes of proving Problem 1.8 with a similar method as Exercise 1.7).
The result can not be true for both odd and even sequences. Otherwise, I mean, if true for both, one could sum up the functions and get $H^1$ function with Fourier coefficients $a_n$ for any convex, non negative, and tending to $0$ sequence $a_n$. However, as it is well known, (Hardy's inequality), for functions in $H^1$ one has $$ \sum \frac{|a_n|}{n} < \infty$$ Also, for any convergent to $0$ sequence of numbers $b_n$ one can find convex, non negative, convergent to zero sequence $a_n$ such that $a_n >= |b_n|$