Book browsing Banach spaces of Analytic function of the author Kenneth Hoffman on page 74 is one example. This example is compiled in this way:
Prove that
$$ \sum_{n=1}^{\infty}\frac{1}{\log n}e^{in\theta} $$
is not the Fourier series of a finite measure on the circle.
I would like to know how is the solution communities in this example. Previously, thank you for solving.
Look at the theorem on p.23. It says that a formal Fourier series is the Fourier series of a finite measure if and only if the Cesaro means are bounded in $L^1$. Check this.