Universal Taylor and trigonometric series in the sense of Menchoff (Q1591465)

A trigonometric series is said to be universal in the sense of Menchoff provided any periodic, measurable function \(f(t)\) exists as the almost everywhere limit of a suitable sequence of partial sums \(\lim_{j\to\infty} S_N(j)(t)\) of the given series. Menchoff proved the existence of universal series in 1945. This note provides a proof that such behavior is typical, that is, for almost every sequence \(\{c_n\}_{n\in\mathbb{N}}\), the series \(\sum_{n\in\mathbb{N}} c_n e^{ i nt}\) is universal. The proof relies on the ability to approximate continuous periodic functions uniformly outside of a subset of the torus having small measure from within a manifold of trigonometric polynomials whose sequences of coefficients are uniformly bounded both in \(\ell^2\) as well as \(\ell^\infty\). The authors go on to verify that a series can be universal even when a fairly strong condition is imposed on the decay of its coefficients. Precisely, suppose that \(\phi\) is a strictly increasing function on \(\mathbb{R}_+\) such that \(\phi(x)=O(x^2)\) as \(x\to 0\), or \(\phi\) is convex and \(\phi(2x)=O(\phi(x))\) as \(x\to 0\). Then there is a universal series \(\sum c_n e^{int}\) such that \(\sum_{n=0}^\infty \phi(|c_n|) <\infty\).