$f$ is said to have trigonometric expansion if some series $\sum_{n\in\mathbb{Z}}c_ne^{inx}$ converges pointwise to $f(x)$. On the second page of the article Trigonometric series and set theory, Alexander S. Kechris says that "even for continuous functions, one can argue that no reasonable characterization (for whether a function has trigonometric expansion) can be found". The author refers this to section (F) of the paper, but as far as I can see, section (F), or indeed the whole paper, is about characterizing sets of uniqueness. Are there any positive or negative solutions to this question? More precisely,

  1. What is the complexity of the set $\{(c_n)_n\mid \sum_{n\in\mathbb{Z}}c_ne^{inx}\text{ converges pointwise}\}$? (the author mentions that we must have $c_n\rightarrow 0$, so this is a subset of the Polish space of sequences converging to $0$)

  2. What is the complexity of the set $\{f\in C(\mathbb{R})\mid f\text{ has a trigonometric expansion}\}$? (such $f$ can be viewed as elements of $C(\mathbb{T})$, again a Polish space)

  3. What is the complexity of the set $\{f\in L^2(\mathbb{T})\mid \text{the Fourier series of $f$ converges pointwise}\}$? (there are trigonometric series that converge pointwise but are not Fourier series, such as $\sum_{n=2}^\infty\frac{\sin nx}{\log n}$)

These questions are related, but may be not exactly the same.



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