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Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable on $[0,2\pi]$.

Let $$S=\lbrace \lbrace a_n\rbrace \in c_0: a_n=\hat{f}(n) \forall n \mbox{ for some function } f\in T\rbrace.$$T\rbrace,$$ where $\hat{f}(n)$ denotes the $n$-th Fourier coefficient of $f$, i.e. $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}\,dx.$$ When I was a graduate student, I was told that no known characterizations of $S$ were known. Is this still true?
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Let $c_0$ be the Banach space of doubly infinite sequences $${ $\lbrace a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 }.$$ \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable on $[0,2\pi]$.

Let $$S={ {a_n}\in $S=\lbrace \lbrace a_n\rbrace \in c_0: a_n=\hat{f}(n) \forall n \mbox{ for some function } f\in T}.$$T\rbrace.$$

When I was a graduate student, I was told that no known characterizations of $S$ were known. Is this still true?
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Characterizations of a linear subspace associated with Fourier series

Let $c_0$ be the Banach space of doubly infinite sequences $${ a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 }.$$ Let $T$ be the space of $2\pi$ periodic functions integrable on $[0,2\pi]$.

Let $$S={ {a_n}\in c_0: a_n=\hat{f}(n) \forall n \mbox{ for some function } f\in T}.$$

When I was a graduate student, I was told that no known characterizations of $S$ were known. Is this still true?