Let $f: \mathbb{R} \longrightarrow \mathbb{C}$ be a periodic function with period $2L>0$, which satisfies the Dirichlet conditions. Then, it's well-defined the Fourier series of $f$ and can be written as
$$
f(t)=\sum_{n \in \mathbb{Z}} c_n \cdot e^{iw_n t}\; dt, \quad t \in [-L,L]
$$
where, for each $n \in \mathbb{Z}$,
$$
c_n:= \frac{1}{2L} \int_{-L}^{L} f(t) \cdot e^{-iw_n t}\; dt \quad \text{and} \quad w_n:= \frac{n \pi}{L}.
$$
Here $(c_n)_{n \in \mathbb{Z}}$ are called Fourier coefficients of $f$.
Question. If we define $g: \mathbb{R} \longrightarrow \mathbb{R}$ by
$$
g(t):= |f(t)|, \; \forall \; t \in \mathbb{R}
$$
then what about the Fourier coefficients of $g$?
If we denote by $(d_n)_{n \in \mathbb{Z}}$ the Fourier coefficients of $g$, then we have
$$
d_n=|c_n|,\quad n \in \mathbb{Z}\;?
$$
By explicit computations, we see that
$$
|c_n| \leq \frac{1}{2L} \int_{-L}^{L} |f(t)|\; dt=\frac{1}{2L} \int_{-L}^{L} g(t)\; dt,\quad n \in \mathbb{Z}.
$$
But, what can I to conclude from this? There is some relation between the Fourier coefficients of $f$ and $g$?
Best Answer
It's certainly not true that $d_n = |c_n|$ in general. A useful example is $f(t) = \cos(t)$ (with $L = \pi$). Then $c_n = 1/2$ for $n = \pm 1$, $0$ otherwise. But $|\cos(t)|$ has infinitely many nonzero Fourier coefficients: in fact I get $d_n = 2 \cos(n\pi/2)/(\pi (1-n^2))$ for $n \ne \pm 1$.