Suppose $\hat{f}(n)$ is the fourier coefficient of a function $f$ on the circle. I have seen in class that for $p=2,$ if $\{\hat{f}(n)\}_{n \mathbb{zZ}}$ is in $\ell^p$, then we have f is in the $L^p$ space.
I.e. if $\sum|\hat{f}(n)|^2< \infty,$ then f is in $L^2$.
Is it also true for p=1?
I.e. is it true that $\sum |\hat{f}(n)|< \infty$, then f is in $L^1?$
If $\sum |\hat{f}(n)|< \infty$, then the Fourier series converges uniformly. Therefore, its sum is a continuous function. A continuous function on a circle is integrable, so yes, the conclusion is true.
However, it's not as useful as the $p=2$ case because it cannot be inverted: there are plenty of $L^1$ functions whose Fourier coefficients are not in $\ell^1$ (for example, any bounded discontinuous function).