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Do I need to use conjugation rules to show that $\overline{h(t)}=\hat{g}(t)$ and when $g(t)=\overline{\hat{h}(t)}$? Trying to prove parsevals identity with this one.

Edit: Something like this: $$\mathcal{F}g(t)=\mathcal{F}\overline{\hat{h}(t)} =\overline{\int_{-\infty}^{\infty}\hat{h}(t)e^{-i2\pi vt}dt}=\int_{-\infty}^{\infty}\overline{\hat{h}(t)}e^{i2\pi vt}dv=\overline{h(t)}$$ Because the complex conjugate makes it the inverse fourier transform for formula? I think I can handle the Parsevals identity if this is the logic behind this.

ELEC
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A notation that might make this easier: $$ g^{\wedge}(s) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}g(t)e^{-ist}\,dt,\;\;\;\; g^{\vee}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}g(t)e^{ist}\,dt. $$ For $L^{2}(\mathbb{R})$, one has $(g^{\wedge})^{\vee}=g$. In general, $\overline{g^{\wedge}}=\overline{g}^{\vee}$ and $\overline{g^{\vee}}=\overline{g}^{\wedge}$. It's a common and convenient notation because the order of conjugation can be swapped if you turn the $\vee$ or $\wedge$ upside down. For example, in your notation, $g=\overline{h^{\wedge}}$, and $$ \mathcal{F}g = g^{\wedge}=(\overline{h^{\wedge}})^{\wedge}=\overline{(h^{\wedge})^{\vee}}=\overline{h}. $$

Disintegrating By Parts
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  • Hmmm we haven't used that kind of notation on our course, but that makes sense. What I wrote also makes sense yeah? – ELEC Jan 17 '14 at 21:27
  • @ELEC: They're the same thing. I kept getting confused by the same things when I started with the transforms until I started using the hat-wedge notation. Just thought I'd pass it along as it was passed along to me. – Disintegrating By Parts Jan 17 '14 at 21:30
  • You are missing a square root in the factor of the inverse transform. Does putting accents on the letters also work? $\overline{\check g}=\hat{\overline g}$, $\overline{\hat g}=\check{\overline g}$. Not really, looks rather small. – Lutz Lehmann Jan 18 '14 at 08:20
  • @LutzL: I corrected the missing square root. Thanks. Yes, going vertical makes complicated expressions too hard to read. – Disintegrating By Parts Jan 18 '14 at 10:31