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

[Clammierfire20]

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!

 

[jvicens]

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!