Universal Taylor series

A universal Taylor series is a formal power series $\sum _{n=1}^{\infty }a_{n}x^{n}$, such that for every continuous function $h$ on $[-1,1]$, if $h(0)=0$, then there exists an increasing sequence $\left(\lambda _{n}\right)$ of positive integers such that

$$\lim _{n\to \infty }\left\|\sum _{k=1}^{\lambda _{n}}a_{k}x^{k}-h(x)\right\|=0$$

In other words, the set of partial sums of $\sum _{n=1}^{\infty }a_{n}x^{n}$ is dense (in sup-norm) in $C[-1,1]_{0}$, the set of continuous functions on $[-1,1]$ that is zero at origin.

Statements and proofs

Fekete proved that a universal Taylor series exists.

Proof

Let $f_{1},f_{2},...$ be the sequence in which each rational-coefficient polynomials with zero constant coefficient appears countably infinitely many times (use the diagonal enumeration). By Weierstrass approximation theorem, it is dense in $C[-1,1]_{0}$. Thus it suffices to approximate the sequence. We construct the power series iteratively as a sequence of polynomials $p_{1},p_{2},...$, such that $p_{n},p_{n+1}$ agrees on the first $n$ coefficients, and $\|f_{n}-p_{n}\|_{\infty }\leq 1/n$.

To start, let $p_{1}=f_{1}$. To construct $p_{n+1}$, replace each $x$ in $f_{n+1}-p_{n}$ by a close enough approximation with lowest degree $\geq n+1$, using the lemma below. Now add this to $p_{n}$.

Lemma — The function $f(x)=x$ can be approximated to arbitrary precision with a polynomial with arbitrarily lowest degree. That is, $\forall \epsilon >0,n\in \{1,2,...\}\exists$ polynomial $p(x)=a_{n}x^{n}+\cdots +a_{N}x^{N},$ such that $\|f-p\|_{\infty }\leq \epsilon$.

Proof of lemma

The function $g(x)=x-c\tanh(x/c)$ is the uniform limit of its Taylor expansion, which starts with degree 3. Also, $\|f-g\|_{\infty }<c$. Thus to $\epsilon$-approximate $f(x)=x$ using a polynomial with lowest degree 3, we do so for $g(x)$ with $c<\epsilon /2$ by truncating its Taylor expansion. Now iterate this construction by plugging in the lowest-degree-3 approximation into the Taylor expansion of $g(x)$, obtaining an approximation of lowest degree 9, 27, 81...