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Suppose $f,g\in L_2(\mathbb{R})$, define $$[f,g](x)=\sum_{k\in\mathbb{Z}}f(x+2\pi k)\overline{g(x+2\pi k)}$$ if the the series converges absolutely and $[f,g](x)=0$ otherwise. Then in this case we see that $[f,g]$ is a $2\pi$ periodic function. Moreover use Tonelli-Fubini theorem and Cauchy-Schwarz inequality we have $$\begin{aligned}\int_0^{2\pi}|[f,g](x)|dx&\leq\sum_{k\in\mathbb{Z}}\int_0^{2\pi}|f(x+2\pi k)||g(x+2\pi k)|dx\\ &=\int_{\mathbb{R}}|f(x)||g(x)|dx\leq \|f\|_{L_2(\mathbb{R})}\|g\|_{L_2(\mathbb{R})}\end{aligned}$$ Thus $[f,g]\in L_1(\mathbb{T})$, and we can calculate its Fourier coefficients, denoted by $$a_n=\int_0^{2\pi}[f,g](x)e^{-inx}dx.$$ Now my question is: In general, can we say anything about the convergence of the Fourier series of $[f,g]$? Moreover I am interested to know whether or not we can conclude that the Fourier series converges to $[f,g]$ almost everywhere.

I know that from the thery of Fourier series, there exist funcitons in $L_1(\mathbb{T})$ such that the Fourier series diverge almost everywhere. But in my question, the function $[f,g]$ is defined somewhat more specific. Any suggestions are appreciated.

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Let $\cal B$ be the class of functions $[f,g]$ with $f,g\in L^2(\mathbb R)$. As $f\bar g\in L^1(\mathbb R)$, we have that $\cal B$ is contained in the class of functions ${\cal C} = \{\sum_{k\in\mathbb Z}f(\cdot + k) : f\in L^1(\mathbb R)\}$. But as each function $f\in L^1(\mathbb R)$ can be represented as $f = g\bar h$ with $g,h\in L^2(\mathbb R)$, we actually have that $\cal B = \cal C$. I will now show that, in fact, ${\cal B} = {\cal C} = L^1(0,1)$. This means that you unfortunately cannot say more about the Fourier coefficients of the functions in $\cal B$.

To show ${\cal C} = L^1(0,1)$, let $F\in L^1(0,1)$ and define $f$ on $\mathbb R$ trivially, namely, $f(x) := F(x)$ for $x\in (0,1)$ and $f(x) = 0$ elsewhere. Then $f\in L^1(\mathbb R)$ and $F(x) := \sum_{k\in\mathbb Z}f(x + k)$ for a.e. $x\in (0,1)$ and so, indeed, $F\in\cal C$.

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