Recently I stumble on the following question when trying to find a bound of a Fourier series-related integral:
Let $\{a_n\}_{n\in\mathbb{Z}}$ be some sequence such that $|a_n|\leq 1$, for all $n$. Let $G(x)$ denote a positive integrable even function defined on $[-\pi,\pi]$. I wonder whether the following holds for all $n$: $$ \int_{-\pi}^{\pi}\sum_{k=-n}^n a_k\cos(kx)G(x)dx =O(1).$$ When $G(x)=1$ is constant, I can evaluate the integral directly to obtain: $$\int_{-\pi}^{\pi}\sum_{k=-n}^n a_k\cos(kx)dx =2\pi a_0+2\sum_{k=1}^n\frac{(a_k+a_{-k})\sin(k\pi)}{k} \leq 2\pi |a_0|\leq 2\pi. $$ I then realized that the quantity of interest could be rewritten as a convolution. Set $f(x) = \sum_{k=-n}^n a_k\cos(kx)$, we have:
$$(f*g)(\tau) = \int_{-\pi}^{\pi}\sum_{k=-n}^n a_k\cos(kx)G(\tau - x)dx.$$ But $f(x)$ is in general unbounded at $x=0$, so can we show that $(f*g)(0)<\infty$ given the conditions here? Some special sequences such as $a_n = 1$ work, but does this result holds for arbitrary sequences?
I tried something like $||f*g||_{L_1}\leq||f||_{L_1} ||g||_{L_1}$ or $||f*g||_{\infty}\leq||f||_{L_1} ||g||_{\infty}$, but $||f||_{L_1}$ diverges as $n\to\infty$, which is not useful.
Any insight or suggestions on this would be greatly appreciated!
Another random thought. Let us write: $$ \int_{-\pi}^{\pi}\sum_{k=-n}^n a_k\cos(kx)G(x)dx = 2\sum_{k=-n}^na_k \int_{-\pi}^{\pi} e^{-ikx}G(x)dx= 4\pi\sum_{k=-n}^na_k\hat{G}[n] $$ where $\hat{G}[n]$ is the $n$th Fourier coefficient of $G$. Suppose $G\in C^p$, we have the estimate $|\hat{G}[n]|\leq K/|n|^p$ for some constant $K$. Then it appears that: $$4\pi\sum_{k=-n}^na_k\hat{G}[n]\leq4\pi\sum_{k=-n}^n\frac{K|a_k|}{|n|^p}\leq 4\pi K\sum_{k=-n}^n\frac{1}{|n|^p},$$ so for the $O(1)$ result to hold, one needs to have $G\in C^p$ for some $p>1$.
If $a_1=-1$, $a_j=0$ for $j\ne1$ and $G=\chi_{[\pi/4,\pi/2]\cup[-\pi/2,-\pi/4]}$ then $$ \int_{-\pi}^{\pi}\sum_{k=-n}^n a_k\cos(kx)G(x)dx > \int_{-\pi}^{\pi}\sum_{k=-n}^n |a_k|\cos(kx)G(x)dx, $$ contradicting the inequality that I thought was the point to the question until it was edited away...
A simpler counterexample to the "$=O(1)$" that remains: If $a_k=1$ for all $k$ and $G_\delta=\delta^{-1}\chi_{[-\delta,\delta]}$ then $$ \sup_{\delta>0}\int_{-\pi}^{\pi}\sum_{k=-n}^n a_k\cos(kx)G_\delta(x)dx =2(2n+1).$$