/sci/ - Science & Math

[Josef crossed the Atlantic !!nUf2NflSAyw No.3931458]

Sup,

I'm looking for a mathematical proof that the inverse Fourier transformation is indeed the inverse of the Fourier transformation. The physicist version is exchanging the integrals because "it works" and then identifying one integral as a delta function, but I can't find a proper proof in any of the books I own.

More accurate description: Let <span class="math">\mathcal F,\overline{\mathcal{F}}\;:~ L^1(\mathbb C,\mathbb C)~\mapsto L^1(\mathbb C,\mathbb C)[/spoiler] be given by<div class="math">
\mathcal F(f)(\omega) = \frac1{\sqrt{2\pi}}\int_{-\infty}^\infty\mathrm dt\;f(t) e^{-\mathrm i\omega t}</div><div class="math">
\overline{\mathcal F}(f)(t) = \frac1{\sqrt{2\pi}}\int_{-\infty}^\infty\mathrm d\omega\;f(\omega) e^{\mathrm i\omega t}</div>Show that<div class="math">\mathcal F\circ\overline{\mathcal F} = \overline{\mathcal F}\circ\mathcal F=\mathrm{Id}_{L^1}</div>

[Josef crossed the Atlantic !!nUf2NflSAyw No.3931526]

bump

(I just noticed I messed the spaces up. Assuming Plancherel, all <span class="math">L^1[/spoiler] should be <span class="math">L^2[/spoiler] I guess.)

[Anonymous No.3931586]

I wish I understood this shit.

[Anonymous No.3931612]

You assume /sci/ is full of something beside pseudo-intellectuals.

[Anonymous No.3931619]

41

[Josef crossed the Atlantic !!nUf2NflSAyw No.3931627]

>>3931612
I've already been pleasantly surprised quite a few times, all it takes is patience. I've got a few more hours to bump this.

[Anonymous No.3931637]

>>3931627
you'll eventually find someone,

in the mean time, enjoy a free bump from someone you've helped.

[Anonymous No.3931643]

>>3931627
>>3931612
I'm studying physics for the sole reason of helping josef, have no worries

[Anonymous No.3931644]

>>3931627
I don't have anything good on me today, but if this p-set can wait, i'll get my Fourier book in the mail in a few days.

[Anonymous No.3931650]

I've had the same experience as OP. Sympathy bump.

[Josef crossed the Atlantic !!nUf2NflSAyw No.3931669]

Fun fact: Proving that <span class="math">\displaystyle\delta(t)=\frac1{2\pi}\int_{-\infty}^\infty\mathrm d\omega\;e^{\mathrm i\omega t}[/spoiler] is usually done by applying the Fourier/inverse Fourier transformation. Hooray, physics! (On a side note, I still don't understand the delta. It's on my todo list at least.)

[Anonymous No.3931682]

Wikipedia is your friend.

http://en.wikipedia.org/wiki/Fourier_inversion_theorem#Proof_of_the_inversion_theorem

[Lion !FvAwVR8cv6 No.3931690]

why do you have it written as 1/sqrt(2pi)?

[Anonymous No.3931711]

yo joesf, whats the o operation? and what is IdL1

[Josef crossed the Atlantic !!nUf2NflSAyw No.3931725]

>>3931682
Oh. Now I feel stupid. Thank you, sir.

>>3931690 why do you have it written as 1/sqrt(2pi)?
Because that makes both transformation nicely symmetric.

>>3931711 yo joesf, whats the o operation? and what is IdL1
<span class="math">\circ[/spoiler] is the composition of functions, <span class="math">(f\circ g)(x) = f\big(g(x)\big)[/spoiler], and <span class="math">\mathrm{Id}_{L^1}[/spoiler] is the identity function on <span class="math">L^1[/spoiler], i.e. <span class="math">\mathrm{Id}_{L^1}(f)=f[/spoiler] for all <span class="math">f\in L^1[/spoiler].