Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$

Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in \ell^{1}(\mathbb Z), \ \text{that is}, \ \sum_{n\in \mathbb Z} |\hat{f}(n)| < \infty \}. $
$A(\mathbb T)$ is normed by
$$\|f\|=\|f\|_{A(\mathbb T)}= \sum_{n\in \mathbb Z} |\hat{f}(n)|; (f\in A(\mathbb T)).$$
Suppose $f_{n} \in A(\mathbb T), \ \|f_{n}\|\leq M,$ (Fix, $M>0$)
$$f(y)= \lim_{n\to \infty} f_{n} (y), (y\in \mathbb T)$$
and $f$ is continuous.

My Question is: Can we expect, $f\in A(\mathbb T)$ and $\|f\|\leq M$ ?

(If answer is positive, and proof is long, proper reference also will be o.k for me;)