Can a function $f:G\to\mathbb{C}$ in $L^p,\ p>1, p\neq 2$ have a Fourier transform $F:\hat{G}\to\mathbb{C}$, where $\hat{G}$ is the Pontryagin dual space of $G$? I believe it can be shown that such a transform exists such that $F$ is in $L^q$, with $1/p+1/q=1$. However, does this not violate Parseval's identity, since $p\neq q\neq 2$?
Violation of Parseval's theorem?
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fourier-analysis