The main topics (issues, problems) of the Fourier transform

Question

To explain what we are looking for, let's have a quick review on some points in Fourier transform on periodic functions in both continuous and discrete cases. We emphasize that our attention is just concerned with the abelian groups $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Z_n}$ and the unit circle $\textbf{T}$.

On $\textbf{T}$, Fourier transform is a $*$-homomorphism from $L^1(\textbf{T})$ to $c_0(\mathbb{Z})$. One of the main question in this context is characterization of some type of functions for which the inversion formula to be hold (point-wisely and uniformly) i.e.,
$$f(x)=\sum \hat{f}(n)e^{inx}.$$ Finding the largest space satisfying the inversion formula is still an open problem. On the other hand, outstanding efforts of Kolomogrov, Kahane, Katznelson and Carleson provide sharp information concerning the points for which the identity $f(x)=\sum \hat{f}(n)e^{inx}$ fails. These two approaches have a long history and are well-organized on some books like Trigonometric series written by Zigmond or Fourier Analysis by Duoandikoetxea.

Q. On the abelian groups $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Z_n}$, What are the main topics of Fourier transform (references ?) for anyone who wants to follow.

Answer 1

Saccone considered the space of all functions on $\mathbb{T}$ with uniformly convergent Fourier series. Call this space $U$. There is a natural norm given by $$\|f\|_U = \sup_{n\in\mathbb{N}} \|S_nf\|_{L^{\infty}}$$ where $S_nf$ is the $n$th partial sum of the Fourier series of $f$. $U$ is complete with this norm.

It was shown in [Saccone2000] that $U$ is not closed under pointwise multiplication of functions, so it's not a uniform algebra. Also, as a Banach space, $U$ has Pelczynski property (V) and Dunford-Pettis property.

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Edit/Correction: $(\exists f,g\in U\hspace{4mm} fg\notin U)$ was not shown in [Saccone2000]. Saccone refers to an article by Kahane & Katznelson in 1965.