- Weierstrass’s theorem states that a continuous function on [0, 1] can be uni formly approximated by polynomials. Can every continuous function on the closed unit disc be approximated uniformly by polynomials in the variable \(z ?\)
Solution
No. Consider the continuous function \[ f(z)=\overline z \] on the closed unit disc.
Suppose there were polynomials \(p_n(z)\) converging uniformly to \(\overline z\) on the closed unit disc. On the positively oriented unit circle \(C\), every polynomial is entire, so Cauchy’s theorem gives \[ \int_C p_n(z)\,dz=0. \] Uniform convergence on \(C\) allows passage to the limit in the contour integral: \[ \int_C \overline z\,dz =\lim_{n\to\infty}\int_C p_n(z)\,dz =0. \tag{1} \] But parametrizing \(C\) by \(z=e^{it}\), \(0\le t\le2\pi\), gives \[ \overline z=e^{-it}, \qquad dz=i e^{it}\,dt, \] and therefore \[ \int_C \overline z\,dz =\int_0^{2\pi} i\,dt =2\pi i\ne0, \] contradicting (1).
Hence not every continuous function on the closed unit disc can be uniformly approximated by polynomials in \(z\).