A non-integrable trigonometric series

Question

I am looking forward to convergent trigonometric series $g(\theta)=\sum_{-\infty}^{+\infty} t_ne^{in\theta}$ ($\forall \theta\in[-\pi,\pi]$) such that $g$ is not Lebesgue integrable!

Answer 1

Here is a way to cook-up trigonometric series that are not (Lebesgue) integrable.

Let $\{c_n:n\in\mathbb{Z}\}\subset\mathbb{R}_+$ be such that both $c_n\searrow0$ and $c_{-n}\searrow0$ as $n\rightarrow\infty$.

Dirichlet's convergence test shows that $$f(x)=\sum_n\operatorname{sign}(n)\,c_ne^{inx}$$ is a pointwise convergent series.

In particular, for $c_n=\frac{1}{\log |n|}\mathbb{1}(|n|\geq2)$ we have that \begin{align*} f(x)=\sum_{|n|\geq2}\frac{e^{inx}}{\operatorname{sign}(n)\log |n|}=2i\sum^\infty_{n=2}\frac{\sin nx}{\log n} \end{align*} is pointwise convergent. However, as $\sum^\infty_{n=2}\frac{1}{n\log n}$ diverges, $f\notin\mathcal{L}_1(\mathbb{S}^1)$.

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Here we are using a well known result about Fourier series:

Theorem: If $f\in\mathcal{L}_1(\mathbb{S}^1)$, then the series $\sum\limits_{n\geq1}\frac{c_n(f)-c_{-n}(f)}{n}$ is convergent.