Universal Taylor series information

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.^[1]

^ Mouze, A.; Nestoridis, V. (2010). "Universality and ultradifferentiable functions: Fekete's theorem". Proceedings of the American Mathematical Society. 138 (11): 3945–3955. doi:10.1090/S0002-9939-10-10380-3. ISSN 0002-9939.

[1] Mouze, A.; Nestoridis, V. (2010). "Universality and ultradifferentiable functions: Fekete's theorem". Proceedings of the American Mathematical Society. 138 (11): 3945–3955. doi:10.1090/S0002-9939-10-10380-3. ISSN 0002-9939.

Universal Taylor series information

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