Setting: Let $A : \mathbb R^2 \to \mathbb R$ be a function with the property that for fixed $y \in \mathbb R$ the restriction $x\mapsto A(x,y)$ is smooth on $\mathbb R$. Further, we have for every $m \in \mathbb Z$ a smooth function $b_m:\mathbb R \to \mathbb R$. There exists a large subset $D \subset \mathbb R$ with $\mathbb R \setminus D$ being of Lebesgue measure zero that satisfies $$A(x,y)=\sum_{m \in \mathbb Z} b_m(x)\exp(2\pi i m y)$$ for all $x \in D$ and $y \in \mathbb R$. This convergences is even absolute.
Question: Does this imply that $$\sum_{m \in \mathbb Z} b_m(x)\exp(2\pi i m y)$$ converges absolutely to $A(x,y)$ for all $x \in \mathbb R$ (and not only for the $x \in D$)?