Universal Taylor series

A universal Taylor series is a formal power series [math]\displaystyle{ \sum_{n=1}^\infty a_n x^n }[/math], such that for every continuous function [math]\displaystyle{ h }[/math] on [math]\displaystyle{ [-1,1] }[/math], if [math]\displaystyle{ h(0)=0 }[/math], then there exists an increasing sequence [math]\displaystyle{ \left(\lambda_n\right) }[/math] of positive integers such that[math]\displaystyle{ \lim_{n\to\infty}\left\|\sum_{k=1}^{\lambda_n} a_k x^k-h(x)\right\| = 0 }[/math]In other words, the set of partial sums of [math]\displaystyle{ \sum_{n=1}^\infty a_n x^n }[/math] is dense (in sup-norm) in [math]\displaystyle{ C[-1,1]_0 }[/math], the set of continuous functions on [math]\displaystyle{ [-1,1] }[/math] that is zero at origin.

Statements and proofs

Fekete proved that a universal Taylor series exists.

Proof

Let [math]\displaystyle{ f_1, f_2, ... }[/math] 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 [math]\displaystyle{ C[-1,1]_0 }[/math]. Thus it suffices to approximate the sequence. We construct the power series iteratively as a sequence of polynomials [math]\displaystyle{ p_1, p_2, ... }[/math], such that [math]\displaystyle{ p_n, p_{n+1} }[/math] agrees on the first [math]\displaystyle{ n }[/math] coefficients, and [math]\displaystyle{ \|f_n - p_n \|_\infty \leq 1/n }[/math].

To start, let [math]\displaystyle{ p_1 = f_1 }[/math]. To construct [math]\displaystyle{ p_{n+1} }[/math], replace each [math]\displaystyle{ x }[/math] in [math]\displaystyle{ f_{n+1} - p_n }[/math] by a close enough approximation with lowest degree [math]\displaystyle{ \geq n+1 }[/math], using the lemma below. Now add this to [math]\displaystyle{ p_n }[/math].

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

Proof of lemma

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