Consider \(L^1(T)\) to be the space of periodic functions on \(\mathbb R\) of period \(2\pi \) which are integrable on \([0,2\pi ]\). Let \(M\) be a proper closed translation invariant subspace of \(L^1(T)\).
(a) Prove that there exists a non-zero \(h\in L^\infty (T)\) such that \(f*h(x)=0\) for all \(f\in M\), \[ \text {where } f*h(x)=\frac {1}{2\pi }\int _0^{2\pi } f(x-y)h(y)\,dy; \]
(b) use part (a) to prove there must be an \(n\in \mathbb Z\) such that \(\hat f(n)=0\) for every \(f\in M\), where \(\hat f(n)\) is the \(n\) th Fourier coefficient of \(f\).