Why is Fourier transform of $f\in L^1(G)$ defined as $\hat{f}(\gamma)=\int f(x) \gamma(x^{-1}) dx$ instead of $\int f(x) \gamma(x) dx$?
Here $G$ is a locally compact abelian group, $\gamma$ is a continuous character on $G$, i.e., a continuous homomorphism that maps G to $\mathbb{T}$ . $\int f(x) \gamma(x) dx$ seems more natural if we view $\gamma$ as a linear functional on $L^1(G)$.
Update, One may define Fourier transform by $\hat{f}(\gamma)=\int f(x)\gamma(x)dx$ or $\hat{f}(\gamma)=\int f(x)\overline{\gamma(x)}dx$ depending on his/her preference. The inversion formula will then be defined accordingly.