Exercise 4 in chapter 9 of real and complex analysis:
Give examples of $f\in L^2$ such that $f\notin L^1$ but $\hat{f}\in L^1$. Under what circumstances can this happen?
I know function $\frac{\sin x}{x}$ satisfied the condition. But I'm curious about the second part. Who can give some suggestion about under what circumstance can this happen?
Thank all of you!
Smoothness of $f$ translates into size conditions for $\hat f$. Thus, $f$ should be smooth. On the other hand, if $f$ is integrable, then $\hat f$ is continuous. Another condition is then that $\hat f$ is discontinuous.