# Inverse Fourier Transform - left and right? [duplicate]

Possible Duplicate:
Surjectivity of the Fourier Transform on Schwartz Space

Consider the Fourier transform on Schwartz space, given by $$\mathcal{F}(f)(\xi)= \hat{f}(\xi) = (2\pi)^{-\frac{1}{2}}\int e^{-i\xi x} f(x) \, \mathrm dx$$

I understand a proof in my notes that shows that we have a left inverse $$\mathcal{F}^{-1}(\hat{f})(x) = (2\pi)^{-\frac{1}{2}}\int e^{i\xi x} \hat{f}(\xi) \, \mathrm d\xi$$

but how do I know that this is also the right inverse?

Many thanks for hints!

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## marked as duplicate by Hans Lundmark, Jennifer Dylan, William, Nate Eldredge, Ｊ. Ｍ.Aug 24 '12 at 13:18

We have for a function $g\in\mathcal S(\mathbb R)$: $\mathcal F(h)(x)=\mathcal F^{-1}(\widetilde h)(x)$ where $\widetilde h(x)=h(-x)$ (it'sjust a substitution). Applying this result to $h(x)=\mathcal F^{-1}(f)(x)$, we have \begin{align*} \mathcal F(\mathcal F^{-1}(f))(x)&=\mathcal F\left(\frac 1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{i\xi x}f(\xi)d\xi\right)\\ &=\mathcal F^{-1}\left(\frac 1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{i\xi -x}f(\xi)d\xi\right)\\ &=\mathcal F^{-1}(\mathcal F(f))(x)\\ &=f(x), \end{align*} since $\mathcal F^{-1}$ is the left-inverse of $\mathcal F$.