Proof check for Plancherel's theorem (Fourier Transform version)

In summary: Keep up the good work! In summary, the individual has presented a proof for Plancherel's theorem for functions in both the $L^1$ and $L^2$ spaces. The proof involves using a different definition of the Fourier transform and justifying the use of the Monotone Convergence Theorem. The individual has also included their own comments and concerns about the proof and has requested feedback. The summary also includes suggestions for improvement and clarification in the original proof.
  • #1
Clammierfire20
1
2
TL;DR Summary
I've constructed a proof for Placherel's theorem but I would really appreciate it if someone could check it, please. I'm not sure I've implemented the convergence result of the convolution correctly.
I'm trying to prove Plancherel's theorem for functions $$f\in L^1\cap L^2(\mathbb{R})$$. I've included below my attempt and I would really appreciate it if someone could check this for me please, and give me any feedback they might have.

**Note:** I am working with a slightly different definition of the Fourier transform to usual, namely:
$$\widehat{f}(u)=\int_{-\infty}^{\infty}e^{iux}f(x)\,dx$$.
My main worries are that I have not implemented the convergence property of the convolution properly and I'm not sure if I've justified the use of the MCT in the correct way.

## My proof goes as follows:

Firstly, I have previously shown that:

$$(G_\lambda\ast f)(x)\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-iux}e^{-\frac{(u/\lambda)^2}{2}}\widehat{f}(u)\,du. \tag{*}$$

Then, writing $|f(x)|=f(x)\overline{f(x)}$, we have:

$$||{f}||_2^2=\int_{-\infty}^{\infty}f(x)\overline{f(x)}\,dx=\lim_{\lambda\to\infty}\int_{-\infty}^{\infty}f(x)\overline{(G_\lambda \ast f)(x)}\,dx.$$The last inequality holds since $$||G_\lambda \ast f-f||_2 \to 0$$ (something I've already proved). Implementing (*) and then applying Fubini's theorem (which I have justified separately), we can write:

$$\begin{align*}
||{f}||_2&=\lim_{\lambda \to \infty}\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x)e^{iux}\overline{\widehat{f}(u)}e^{-\frac{(u/\lambda)^2}{2}}\,du\,dx\\
&=\lim_{\lambda \to \infty}\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x)e^{iux}\,dx\overline{\widehat{f}(u)}e^{-\frac{(u/\lambda)^2}{2}}\,du\\
&=\lim_{\lambda \to \infty}\frac{1}{2\pi}\int_{-\infty}^{\infty}\widehat{f}(u)\overline{\widehat{f}(u)}e^{-\frac{(u/\lambda)^2}{2}}\,du\\
&=\lim_{\lambda \to \infty}\frac{1}{2\pi}\int_{-\infty}^{\infty}|\widehat{f}(u)|^2e^{-\frac{(u/\lambda)^2}{2}}\,du.
\end{align*}$$

Define the sequence $$\{x_\lambda\}_{\lambda =1}^\infty$$ by $$x_\lambda=|\widehat{f}(u)|^2e^{\frac{-(u/\lambda)^2}{2}}$$. Clearly, we have $$\lim_{\lambda\to\infty}x_n=|\widehat{f}(u)|$$. We also know $\{x_n\}$ is non-negative and monotone increasing, since $|\widehat{f}(u)|$ is fixed and $e^{\frac{-(u/\lambda)^2}{2}}$ increases in value as $\lambda$ increases. Therefore, by the MCT, we have:

$$||{f}||_2=\frac{1}{2\pi}\int_{-\infty}^{\infty}|\widehat{f}(u)|^2\lim_{\lambda \to\infty}e^{-\frac{(u/\lambda)^2}{2}}\,du=\frac{1}{2\pi}\int_{-\infty}^{\infty}|\widehat{f}(u)|^2\,du=\frac{1}{2\pi}||{\widehat{f}}||_2^2.$$

Any help is really appreciated, thank you in advance!
 
  • Like
Likes nuuskur and Delta2
Physics news on Phys.org
  • #2


Hello! Your proof looks good overall. I have a few comments and suggestions:

1. In the first line of your proof, you have written $(G_\lambda\ast f)(x)$ instead of $G_\lambda \ast f(x)$. It's a small typo, but it's important to be consistent with your notation throughout the proof.

2. In the second line, you have written $||{f}||_2^2$ instead of $||f||_2^2$. Again, a small typo, but it's important to be consistent.

3. In the fourth line, you have written "The last inequality holds since..." but it's not clear what inequality you're referring to. It would be helpful to specify which inequality you're using.

4. In the fifth line, you have written "something I've already proved". It would be helpful to include a reference or a brief explanation of where you proved this.

5. In the sixth line, you have used Fubini's theorem, which is a great choice. However, it would be helpful to specify which version of Fubini's theorem you're using (e.g. Tonelli's theorem or Fubini's theorem for Lebesgue integrals).

6. In the last line, you have written "Define the sequence $\{x_\lambda\}_{\lambda =1}^\infty$ by..." but it's not clear why you're defining this sequence. It would be helpful to explain the motivation behind this definition.

7. In the last line, you have used the Monotone Convergence Theorem (MCT), which is correct. However, it would be helpful to state the MCT explicitly and explain why it applies in this situation.

Overall, your proof is well-written and easy to follow. I hope my suggestions were helpful and good luck with your research!
 

Related to Proof check for Plancherel's theorem (Fourier Transform version)

1. What is Plancherel's theorem?

Plancherel's theorem is a mathematical theorem that states that the integral of the squared absolute value of a function in the time domain is equal to the integral of the squared absolute value of its Fourier transform. It is an important tool in the field of Fourier analysis and has many applications in signal processing, image processing, and quantum mechanics.

2. What is the Fourier transform version of Plancherel's theorem?

The Fourier transform version of Plancherel's theorem is a variation of the original theorem that applies specifically to the Fourier transform. It states that the integral of the squared absolute value of a function in the time domain is equal to the integral of the squared absolute value of its Fourier transform, multiplied by a constant scaling factor.

3. How is Plancherel's theorem used in signal processing?

In signal processing, Plancherel's theorem is used to analyze signals in the frequency domain. By taking the Fourier transform of a signal and applying Plancherel's theorem, we can determine the power spectrum of the signal, which can provide valuable information about its frequency components and overall characteristics.

4. What is the significance of Plancherel's theorem in quantum mechanics?

In quantum mechanics, Plancherel's theorem is used to relate the wave function of a particle in position space to its wave function in momentum space. This allows for a better understanding of the behavior of particles at the quantum level and is an important tool in studying quantum systems and phenomena.

5. Can Plancherel's theorem be applied to any type of function?

Yes, Plancherel's theorem can be applied to any function that satisfies certain conditions, such as being square-integrable. However, the theorem is most commonly used for functions that have a Fourier transform, such as periodic functions or functions with finite energy.

Similar threads

Replies
4
Views
509
Replies
2
Views
545
  • Topology and Analysis
Replies
4
Views
486
  • Topology and Analysis
Replies
5
Views
2K
  • Topology and Analysis
Replies
2
Views
2K
  • Topology and Analysis
Replies
1
Views
575
Replies
4
Views
545
  • Topology and Analysis
Replies
4
Views
1K
  • Topology and Analysis
Replies
2
Views
1K
Replies
1
Views
1K
Back
Top