414/514 Homework 3 – Continuity and series

November 11, 2014

This set is due in three weeks, on Friday, December 5th, at the beginning of lecture.

1. Given a finite set A\subseteq\mathbb T=\{z\in\mathbb C\mid |z|=1\}, find (with proof) a power series f(x)=\sum_{n=0}^\infty a_n x^n with radius of convergence 1 and such that if z\in\mathbb T, then f(z) converges iff z\notin A.

Find (with proof) a countably infinite set  X\subseteq \mathbb T and a power series f(x)=\sum_{n=0}^\infty a_n x^n with radius of convergence 1 and such that if z\in\mathbb T, then f(z) converges iff z\notin X.

The first part is easy, the second one may be tricky. If after a few honest efforts you find yourself stuck, take a look at the literature. For instance, at either Stefan Mazurkiewicz, Sur les séries de puissances, Fundamenta Mathematicae, 3, (1922), 52–58, or Fritz Herzog, George Piranian, Sets of convergence of Taylor series. I. Duke Math. J., 16, (1949), 529–534. Both papers prove more general results, by explicit constructions. Present a detailed adaptation of their argument that solves the question. Do not simply reproduce the proof in the papers for the more general cases. Other relevant references on this topic are Fritz Herzog, George Piranian, Sets of convergence of Taylor series. II. Duke Math. J., 20, (1953), 41–54, and Thomas W. Körner, The behavior of power series on their circle of convergence. In Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981), 56–94, Lecture Notes in Math., 995, Springer, Berlin-New York, 1983.

2. Present a detailed proof of Weierstrass theorem that for any compact interval [a,b], any continuous function f:[a,b]\to\mathbb R is the uniform limit of a sequence of polynomials. The proof should be different from the one presented in lecture (based on Bernstein polynomials). Again, although many books have a proof of this result, make sure your write up is your own.