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. $$

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