Analysis Problem of the Day 49

Today’s question is Problem 12 from the UCLA Spring 2008 Analysis Qual:

Problem 12: Show that the function $f(z) = \frac{1}{z}$ cannot be uniformly approximated by polynomials on any open annulus $A=\{0<r<|z|<R\}.$

Solution: If $f(z)$ is approximated uniformly by polynomials $f_n$ on $A,$ then $z f_n(z)-1$ is a sequence of polynomials converging to $0$ uniformly on $A.$ By the maximum modulus principle, it follows that $z f_n(z)-1$ converges uniformly to 0 on $\{|z| \leq R\},$ which implies that $z f_n(z) \to 1$ uniformly on $\{0<|z|<R\}.$ But this is impossible since $z f_n(z)=0$ at $z=0$ for all $n.$ Thus, such a sequence of polynomials does not exist.

Remark: Notice that the following argument shows that no meromorphic function can be approximated uniformly by polynomials on any annulus around a singularity.

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