Universal Taylor series

A universal Taylor series is a formal power series $\sum _{n=1}^{\mathrm{\infty }}{a}_n{x}^n$, 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$$\underset{n\to \mathrm{\infty }}{\lim }\Vert \sum _{k=1}^{{\lambda }_n}{a}_k{x}^k-h(x)\Vert =0$$In other words, the set of partial sums of $\sum _{n=1}^{\mathrm{\infty }}{a}_n{x}^n$ 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 $\Vert f_n-p_n{\Vert }_{\mathrm{\infty }}\le 1/n$.

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 $\ge n+1$, 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, $\mathrm{\forall }ϵ>0,n\in \{1,2,...\}\mathrm{\exists }$ polynomial $p(x)=a_n{x}^n+\cdots +a_N{x}^N,$ such that $\Vert f-p{\Vert }_{\mathrm{\infty }}\le ϵ$.

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, $\Vert f-g{\Vert }_{\mathrm{\infty }}<c$. Thus to $ϵ$-approximate $f(x)=x$ using a polynomial with lowest degree 3, we do so for $g(x)$ with $c<ϵ/2$ 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...