$f(t)=\sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}$ for $f\in L^{2}(\mathbb T)$?

Question

Let $f\in L^{2}(\mathbb T).$ Define $g(t):= \sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}, (t\in \mathbb T).$

Since $\hat{f} \in \ell^{2}(\mathbb Z),$ we note that $g\in L^{2}(\mathbb T).$

My Question is: Can we say that: $f(t)=g(t)$ for all $t\in \mathbb T$ (pointwise equality) ? If yes, how to prove it? If not, any counter example?