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
[edit]Fekete proved that a universal Taylor series exists.[2]
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 .
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
[edit]- ^ 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.
- ^ Pál, Julius (1914). "Zwei kleine Bemerkungen". Tohoku Mathematical Journal. First Series. 6: 42–43.