Universal Taylor series

A universal Taylor series is a formal power series , such that for every continuous function on , if , then there exists an increasing sequence of positive integers such thatIn other words, the set of partial sums of is dense (in sup-norm) in , the set of continuous functions on that is zero at origin.[1]

Statements and proofs

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Fekete proved that a universal Taylor series exists.[2]

Proof

Let   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  . Thus it suffices to approximate the sequence. We construct the power series iteratively as a sequence of polynomials  , such that   agrees on the first   coefficients, and  .

To start, let  . To construct  , replace each   in   by a close enough approximation with lowest degree  , using the lemma below. Now add this to  .

Lemma — The function   can be approximated to arbitrary precision with a polynomial with arbitrarily lowest degree. That is,   polynomial   such that  .

Proof of lemma

The function   is the uniform limit of its Taylor expansion, which starts with degree 3. Also,  . Thus to  -approximate   using a polynomial with lowest degree 3, we do so for   with   by truncating its Taylor expansion. Now iterate this construction by plugging in the lowest-degree-3 approximation into the Taylor expansion of  , obtaining an approximation of lowest degree 9, 27, 81...

References

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  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.
  2. ^ Pál, Julius (1914). "Zwei kleine Bemerkungen". Tohoku Mathematical Journal. First Series. 6: 42–43.