Prove that trigonometric series is/isn't a Fourier series

Question

Prove that trigonometric series $\sum_{n=2}^{\infty} \frac{\cos(nx)}{\ln(n)}$ is a Fourier series and $\sum_{n=2}^{\infty} \frac{\sin(nx)}{\ln(n)}$ is not.

I know a few tricks how to find out whether the trigonometric series are Fourier or not: check its uniform convergence, check the weights convergence to zero; check the Bessel's inequality for the weights.

But these series look so similar - I'm at a loss.

Answer 1

Theorem 4.2 of "Introduction to Harmonic Analysis" (Y. Katznelson) states that

Let $f\in L^1(\mathbb{T})$ and assume that $\hat{f}(|n|)=-\hat{f}(-|n|)\geq 0$. Then $$\sum_{n>0}\frac{\hat{f}(n)}{n}<\infty$$

As a corollary, if $a_n>0$ and $\sum_n a_n/n$ is not finite, then the sine series $\sum_{n}a_n\sin nt $ is not a Fourier series.

Answer 2

This question is fully answered in example-of-a-trigonometric-series-that-is-not-fourier-series?rq=1. Look at the second answer in that page.

In summary, by the Dirichlet convergence test, the sine series above converges everywhere and the cosine series converges everywhere except for integer multiples of $2\pi$. [Indeed the Dirichlet convergence test shows that they converge uniformly on any compact set which excludes integer multiples of $2\pi$.] However by the criterion cited above in Katznelson's book (pages 24 & 73), the sine series converges to a function which is not in $L^1$. By definition the constant term coefficient of a Fourier series is the Lebesgue integral $$c_0=\frac{1}{2\pi}\int_{-\pi}^\pi f(\theta)\ d\theta,$$ which is only defined when $f\in L^1$. Hence the sine series is not a Fourier series. Katznelson (pages 23-24) also shows that the cosine series converges to a function in $L^1$ and that the cosine series is the Fourier series of that function.