Plancherel's Theorem proof

In summary, the first step of the Plancherel's Theorem proof involves using the definition of Fourier Transform pairs and taking the complex conjugate of the first equation. This is done by changing the variable in the integral and using the fact that definite integrals are unaffected by variable changes.
  • #1
the first step of the Plancherel's Theorem proof found in: http://mathworld.wolfram.com/PlancherelsTheorem.html, says:
let
Inline1.gif
be a function that is sufficiently smooth and that decays sufficiently quickly near infinity so that its integrals exist. Further, let
Inline2.gif
and
Inline3.gif
be FT pairs so that:
Inline4.gif
Inline5.gif
Inline6.gif

Inline7.gif
Inline8.gif
Inline9.gif


assuming x = 2*pi*v*t, why is E(t) multiplied by e^(-ix)?, i guess it has to do with the fact that it is the conjugate of e^(ix), but i can't figure it out
 
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  • #2
The first equation follows from the definition of Fourier Transform pairs (to be more precise from a theorem that the inverse Fourier transform of the Fourier transform of a function, is the function itself).

The second equation follows from first by taking the complex conjugate at each side of the first equation. And of course changing the name of the variable but I guess you know that ##\int f(x)dx=\int f(y)dy## no matter what x and y are.
 
  • #3
but if we have for example:
f(x) = x³
y = x²
then
f(y) = (x²)³ = x⁶
dy/dx = 2x
dy = 2x dx

using the equation you suggest:
∫f(x)dx=∫f(y)dy
∫x³ dx=∫2x⁷ dx

i'm missing something?
 
  • #4
Well the equation I wrote is for definite integrals (ok I admit I didn't write it in an accurate way) , so i should ve write ##\int\limits_{a}^{b}f(x)dx=\int\limits_{a}^{b}f(y)dy##

What you doing is a change of variable ##y=x^2## in the integral ##\int\limits_{a}^{b}f(y)dy## so the interval of integration changes from ##(a,b)## to ## (\sqrt{a},\sqrt{b})##. So the last line of your post should be actually ##\int\limits_{a}^{b}x^3dx=\int\limits_{\sqrt{a}}^{\sqrt{b}}2x^7dx##.
 
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What is Plancherel's Theorem?

Plancherel's Theorem is a mathematical theorem that relates the Fourier transform of a function to its square-integral. It is named after French mathematician Michel Plancherel.

What does Plancherel's Theorem state?

Plancherel's Theorem states that the integral of the squared absolute value of a function in one domain is equal to the integral of the squared absolute value of its Fourier transform in the other domain.

What is the significance of Plancherel's Theorem?

Plancherel's Theorem is a fundamental result in Fourier analysis and is used in various fields of mathematics, physics, and engineering. It allows for the conversion between functions in the time and frequency domains, making it a powerful tool for studying and analyzing signals and systems.

What is the proof of Plancherel's Theorem?

The proof of Plancherel's Theorem involves using the properties of the Fourier transform, such as linearity, duality, and the convolution theorem. It also uses techniques from measure theory and complex analysis. The complete proof can be found in most advanced mathematical analysis textbooks.

Are there any variations of Plancherel's Theorem?

Yes, there are variations of Plancherel's Theorem for different types of Fourier transforms, such as the discrete-time Fourier transform and the discrete Fourier transform. These variations have similar principles but are tailored to different types of signals and systems.

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