$f$ is real valued iff $\overline{ \hat{f}(n) } = \hat{f}(-n)$

Question

The problem I am considering is:

For $f$ a $2\pi$-periodic and Riemann integrable function, show that $f$ is real valued iff $\overline{ \hat{f}(n) } = \hat{f}(-n)$.

Here $\hat{f}(n)$ represents the $n$th fourier coefficient.

It is easy to see that $f$ real implies the equality.

Going the other direction, all I am able to deduce is that $\widehat{\bar{f} - f}(n) = 0$ $\forall n \in \mathbb{Z}$, which suggests that $f$ is real whenever it is continuous. (Since the Fejer kernel is a good kernel, the Fourier series is Cesaro summable to every point of continuity of $f$.)

I know that I can approximate in $L_1$ norm an integrable function on the disc. However, all this would give (provided that I could translate the equality in terms of the approximating sequence) is that $f$ is real a.e. This also seems to be a dead end.

Can someone suggest an alternative approach?

(I am reading Stein and Shakarachi - Fourier Analysis.)

In the next chapter they discuss the convergence of the partial sums in $L^2$, but I do not think I am allowed to use that. (Again, that would only provide real valued a.e.)

Answer 1

Strictly speaking, this isn't true as stated. Let $f$ be $0$ everywhere, except $f(2\pi k)=i$. Then $f$ is Riemann integrable, but the Riemann integral won't pick up the non-real value at the single point, so $\hat f(n)=0$ for all $n$. Real-valued almost everywhere is the best you can hope for.

Answer 2

In part a) of the same question (Fourier Analysis, Stein and Shakiarchi) it is asked to show \begin{equation*} f(\theta) = \hat{f(0)} + \Sigma[\hat{f(n)}+\hat{f(-n)}]cosn\theta + i[\hat{f(n)}-\hat{f(-n)}]sinn\theta \end{equation*} Now let $\hat{f(n)}\ =\ a+ib$ and $\hat{f(-n)}\ =\ c+id$, so according to the given condition $a-c=i(b+d)$.

Therefore $\hat{f(n)}+\hat{f(-n)}\ =\ (a+c)+i(b+d)\ =\ 2a$ and $\hat{f(n)}-\hat{f(-n)}\ =\ (a-c)+i(b-d)=2ib$. Now put these values in the equation found in part a) and that will yield that $f$ is a real valued function.