If $\hat{f}(k)=0$ for all $k <0$, then $f(x)\geq0$ for all $x$

Question

I just started learning about the Fourier series, is this statement true or false?

Looking at $\mathcal {R}(-\pi,\pi).$ If $\hat{f}(k)=0$ for all $k <0$, then $f(x)\geq0$ for all $x$.

Answer 1

Notice that $\widehat{f}(k)=\frac{1}{2\pi}\int^\pi_{-\pi}e^{-iky}f(y)\,dy=\overline{\widehat{f(-k)}}$

So if $f$ is real, your condition implies that $\widehat{f}(k)=0$ for all $k$ and so $f=f(0)=\widehat{f}(0)$ a.s. So, unless $f(0)\geq0$, the answer to your question is general is NO.