Let $T$ be the unit circle and $D$ the open unit disk. A function $f$ belongs to $C(T)$ if it is continuous at $T$. A function $g$ belongs to $A(D)$ if it is continuous at $\overline{D}$ and holomorphic on $D$. Thus $A(D)|_{T}\subset C(T)$. Now, if we think of the symmetric partial sums of the Fourier series of $f\in C(T)$, we notice that $$\sum_{k=-n}^n a_k(f)e^{ikt}=\sum_{k=0}^n a_k(f)e^{ikt}+\sum_{k=-n}^{-1} a_k(f)e^{ikt}=g_{n}(e^{it})+h_{n}(e^{it}).$$ If we are lucky enough and $$\sum_{k=0}^n a_k(f)e^{ikt}$$ converges uniformly, then the limit will belong to $A(D)$. But this is not always the case. So, I was wondering if $A(D)$ is a topological complement of $C(T)$. I think the answer is negative. Any ideas?
Also, are there any sufficient coditions except from the absolute convergence of the Fourier series of $f$, in order for the sequences $g_{n},h_{n}$ to converge uniformly?