This question is somewhat similar to [Minimizing the L1 norm of odd-term trigonometric polynomial](https://mathoverflow.net/questions/260077/minimizing-the-l1-norm-of-odd-term-trigonometric-polynomial). The context of the question is based on the paper [Hardy's Inequality and the $L^1$ norm of Exponential Sums](https://www.jstor.org/stable/2007000). In that paper, they prove the inequality that given any increasing sequence $n\_1, \dots, n\_k$ of integers,
$$
\left\|\sum a\_j e^{2\pi i n\_j x}\right\|\_1 \geq C \sum \frac{|a\_j|}{j}
$$ for some absolute constant $C$.


A key idea of the paper is that they are able to construct a function $F$ such that


1. $\|F\|\_{\infty} \leq 1$.
2. $\hat F(n\_j) = \Theta(1/j)$ (This is the big-theta notation)


Can one extend this idea to give the same/worse bound for functions of the form $\sum a\_j \cos(2\pi n\_j x)$ where the $n\_j$'s to be positive integers? Or perhaps is there a simpler way by saying something like
$$
\left\|\sum a\_j e^{2\pi i n\_j x}\right\|\_1 \leq C(k) \left\|\sum a\_j \cos(2 \pi i n\_j)\right\|\_1
$$ for some constant $C(k)$ that's not too big?


Concretely, can one prove:
$$
\left\|\sum a\_j \cos(2 \pi n\_j x)\right\|\_1 \geq C \sum \frac{|a\_j|}{j}
R$$ for some absolute constant $C$? Or maybe a worser bound?



A lower bound for the $L^1$ norm of real trigonometric polynomials
This is not true.


Consider the [Fejer Kernel](https://en.wikipedia.org/wiki/Fej%C3%A9r_kernel) $F\_n$. From the explicit formula for $F\_n$ we see that $$\|F\_n\|\_{L^1} \leq C $$ independent of $n$.


On the other hand since $ F\_n = 2 \sum\_{0\leq k \leq n-1} \left(1 - \frac{|k|}{n} \right) \cos(k x)$ we have that $$ \sum\_{0\leq k \leq n-1} \frac{1}{k} \left(1 - \frac{|k|}{n} \right) \sim \log n.$$


