Fourier Transform and conjugations

Question

Consider a sufficiently regular and decaying function $f\in \mathcal{S}(\mathbb{R})$ and define the Fourier transform as $$ \mathcal{F}(f)(\xi)=\int e^{-ix\xi}f(x)dx \quad \hbox{ and } \quad \mathcal{F}^{-1}(f)(x)=\int e^{ix\xi}f(\xi)d\xi. $$ With these definitions I would like to calculate $\mathcal{F}_{\mu\mapsto\mu'}^{-1}[\overline{\widehat{f}(\xi-\mu)}]$, where $\overline{\cdot}$ stands for the complex conjugate. First, using that $\mathcal{F}(\overline{f})(\xi)=\overline{\mathcal{F}(f)(-\xi)}$, we can directly calculate the above quantity, $$ \mathcal{F}_{\mu\mapsto\mu'}^{-1}[\overline{\widehat{f}(\xi-\mu)}]=\mathcal{F}_{\mu\mapsto\mu'}^{-1}[\widehat{\overline{f}}(\mu-\xi)]=e^{i\xi\mu'}\overline{f}(\mu'). $$ However, if instead I tried to verify my result by direct calculations, taking out of the integral the complex conjugate, for some reason I obtained an extra minus sign: $$ \mathcal{F}_{\mu\mapsto\mu'}^{-1}[\overline{\widehat{f}(\xi-\mu)}]=\overline{\int e^{-i\mu\mu'}\widehat{f}(\xi-\mu)d\mu} $$ Then, if I change variables $\widetilde{\mu}=\xi-\mu$, the differential makes appear an extra minus sign, from where we obtain that the last line equals $$ -\overline{\int e^{-i(\xi-\widetilde{\mu})\mu'}\widehat{f}(\widetilde{\mu})d\widetilde{\mu}}=-\overline{e^{-i\xi\mu'}f(\mu')}=-e^{i\xi\mu'}\overline{f}(\mu') $$ which is the same result as before but with an extra negative sign. Does anybody know what am I doing wrong?

Answer 1

Think about convolution $u\ast v(x)=\int u(x-y)v(y)dy$, using change of variable to check why $\int u(x-y)v(y)dy=\int u(y)v(x-y)dy$.

In your case $u(x)=\hat f(x)$ and $v(x)=e^{-ix\mu'}$.

More precisely, when we take reflection in change of variable we have $$\int_{-\infty}^{+\infty} u(-t)dt=\int_{+\infty}^{-\infty}u(s)(-ds)=\int_{-\infty}^{+\infty}u(s)ds.$$

Btw your Fourier inversion is incorrect. You should either multiplying $(2\pi )^{-n/2}$ on both $\mathcal F$ and $\mathcal F^{-1}$, or use $\mathcal F^{-1}f(x)=\frac1{(2\pi)^n}\int f(\xi)e^{ix\xi}d\xi$. Also you should avoid using $\mu\to\mu'$ where we tends to use different symbols between the frequence side and the physical side.