Coefficients of function on finite abelian group

Question

Let $B \subseteq A$ two finite abelian groups. Moreover, let $f\in k^A$ be a function on $A$, s.t. $$ \sum_{a \in A}\, f(a)\xi([a]) =0$$ for all characters $\xi \in \widehat{A/B}$. Does this imply $f(a)=0$ for all $a \notin B$?

Answer 1

That's not true. If $k=\mathbb{C}$ we can take $A=\mathbb{Z}/4\mathbb{Z}=\{0,1,2,3\}$ and $B=\mathbb{Z}/2\mathbb{Z}=\{0,2\}$. The group $A/B$ has only two elements ($\mathbb{Z}/2\mathbb{Z}$ and $1+\mathbb{Z}/2\mathbb{Z}$)

So the assumption gives for the trivial character that $f(0)+f(1)+f(2)+f(3)=0$ and for the other character that $f(0)-f(1)+f(2)-f(3)=0$

This clearly does not mean that $f(1),f(3)=0$. For example if we take $f(0)=1,f(1)=-1,f(2)=-1,f(3)=1$