Let $f$ be a continuous function on the unit circle $T=\{|z|=1\}$. Show that $f$ can be approximated uniformly on $T$ by a sequence of the polynomial in z if and only if $f$ has an extension $F$ that is continuous on the closed unit disk.
Can anyone suggest to me how I do the approximation of complex function?
Let $0<r<1$ and define $F_r(z)=F(rz)$, for $|z| \le 1$.
Since $F$ is analytic in the interior of the unit disk, it can be uniformly approximated on compact subsets by polynomials in $z$, in particular, on $\overline{D_r}(0)$. And since $F$ is continuous on the closed unit disk, it is also uniformly continuous there, so $F_r \to F$ uniformly on the closed unit disk as $r \to 1^-$. Glueing these two pieces should give the desired approximation.