Can we determine whether $f\in L^{p}$ or not ; if we know $\hat{f}$

Question

Let $a_{n}:=\frac{1}{n}$ for all $n\in \mathbb Z\setminus \{0\}$ and $a_{0}= c$ where $c$ is some constant.

Clearly, $a_{n}\in \ell^{2}(\mathbb Z)$, that is, $\sum_{n\in \mathbb Z} |a_{n}|^{2}< \infty.$

We define the function $$f(t):= \sum_{n\in \mathbb Z} a_{n} e^{int}, (e^{it}\in \mathbb T, \ t\in \mathbb R)$$

By the Riesz-Fischer theorem, it follows that $f\in L^{2}(\mathbb T).$ We also note that $L^{p}(\mathbb T) \subset L^{2}(\mathbb T), (p>2).$

My Question: (1) Is it true that $f\in L^{p}(\mathbb T), (p>2)$?

(2) If we know $\hat{f} \in \ell^{2}(\mathbb Z);$ in which situation one can expect $f\in L^{p}(\mathbb T)(p>2)$? (Of course, in general one can not expect this, for instance there exist $f\in L^{2}$ which is not in $L^{p}$)

Answer 1

Since $\hat{f}\in\ell^1\implies f\in L^\infty$ and $\hat{f}\in\ell^2\implies f\in L^2$, Riesz-Thorin interpolation guarantees that

If $\hat{f}\in\ell^q$ where $\frac1p+\frac1q=1$ and $1\le q\le2$, then $f\in L^p$

Therefore, since the $\hat{f}$ you give above is in $\ell^q$ for all $q\gt1$, we have that $f\in L^p(\mathbb{T})$ for all $p\lt\infty$. However, it may not be in $L^\infty(\mathbb{T})$. In fact it won't since the series for $f(0)$ diverges.