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I'm not able to understand what conjugate of a transform means. In my textbook it says that the fourier inverse transform is just the conjugate, $\mathcal{F}^*\mathcal{F} = I$. Though I'm not sure how I can use this...

suppose I'm given $\mathcal{F}[x(t)] = X(\omega)$, how can we calculate $\mathcal{F}^{-1}[x(\omega)]$ using this?

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  • $\begingroup$ Are you sure you've read exactly this statement in your book? $\endgroup$
    – alexjo
    May 13 '16 at 10:17
  • $\begingroup$ After defining the fourier inverse transform in terms of integrals: "As the kernels of these integrals are complex conjugates of each other we have $\mathcal{F}^{−1}=\mathcal{F}^∗, \mathcal{F}^∗ \mathcal{F}=I$ where * denotes the complex conjugate and $I$ is the identity operator" pg. 100 of Advanced Topics in Applied Mathematics For engineering and the physical sciences by Sudhakar Nair $\endgroup$
    – B2VSi
    May 13 '16 at 12:04
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In operatorial sense we have the Fourier integral operator $$ (\mathcal Ff)(x)=\int_{-\infty}^\infty f(y)\,k(x,y) \,\mathrm dy=\hat f(x) $$ where the kernel, according to the sign convention of your book, is $$k(x,y)=\frac{\mathrm e^{ixy}}{\sqrt{2\pi}}$$ The inverse Fourier integral operator is then $$ (\mathcal F^{-1}\hat f)(x)= f(x)=\int_{-\infty}^\infty \hat f(y)\,k^*(x,y) \,\mathrm dy=(\mathcal F^*\hat f)(x)=(\mathcal F^*\mathcal F f)(x) $$ and then $\mathcal F^{-1}=\mathcal F^{*}$ and $\mathcal F^{*}\mathcal F=\mathcal I$.

Actually this is not useful to find the inverse (pay attention on the variables). This is also a consequence of the duality property of Fourier transform usually written $$ X(\omega)=\mathcal F\{x(t)\}\quad\Longrightarrow \quad\mathcal F\{X(t)\}=x(-\omega) $$ so that \begin{align} \mathcal F^*\{X(t)\}=\mathcal F^*\{\mathcal F\{x(\omega)\}\}&=\left(\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}X(t)\mathrm e^{i\omega t}\,\mathrm d t\right)^*\\ &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}X^*(-t)\mathrm e^{i\omega t}\,\mathrm d t\\ &=\mathcal F\{X^*(-t)\}=x(\omega) \end{align} that is $\mathcal F^*\{\mathcal F\{x(\omega)\}\}=x(\omega)$, or $\mathcal F^{*}\mathcal F=\mathcal I$.

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