I'm trying to prove that it exists a non-trivial function $f\in L^1([-\pi,\pi])$ such that the convolution $(F_N*f)(x)=0$ for all x. I've tried to calculate the convolution: $$(F_N*f)(x)=\frac{1}{2\pi} \int_{-\pi}^\pi F_n(x-t)f(t)dt=\frac{1}{2\pi} \int_{-\pi}^\pi \frac{1}{N+1} \left(\frac{\sin(\frac{(N+1)(x-t) }{2})}{\sin(\frac{(x-t)}{2})}\right)^2 f(t)dt.$$ Because the Fejér kernel is even and the integral is symmetric, my conclusion is that the function $f$ must be zero almost everywhere. But I don't understand how I find a function that is non-trivial.
Any help would be appreciated.
Take $f(x)=e^{i(N+1)x}$. Use the formual $F_N(x)=\sum_{|n| \leq N} c_n e^{inx}$ where $c_n=1-\frac {|n|} {N+1}$ to show that $F_N*f=0$.