Parseval's theorem and Fourier Transform proof

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In summary, the conversation discusses Plancherel's theorem and its relation to Parseval's identity, which states that taking Fourier series gives an isometry from ##L^2(S^1)## to ##\ell^2##, and Plancherel's theorem, which states that the Fourier transform gives a self-isometry of ##L^2(\mathbb{R})\cap L^1(\mathbb{R})##. The proof of the latter involves the use of the Dirac delta.
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TL;DR Summary
I am searching for a proof that the square of a function is equal to the sum of the square of its transform.
Given a function F(t)
$$F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega$$
I am looking for a proof of the following:

$$\int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d \omega$$

Last edited:
The idea is similar. Parseval's identity says that taking Fourier series gives an isometry ##L^2(S^1)\to\ell^2##. Plancherel says that the Fourier transform gives a self-isometry of ##L^2(\mathbb{R})\cap L^1(\mathbb{R})##.

1. What is Parseval's theorem?

Parseval's theorem is a mathematical theorem that relates the energy or power of a signal in the time domain to its frequency domain representation. It states that the total energy or power of a signal in the time domain is equal to the sum of the squared magnitudes of its Fourier transform in the frequency domain.

2. What is the significance of Parseval's theorem?

Parseval's theorem is significant because it allows us to analyze signals in both the time and frequency domains interchangeably. This is useful in many fields, including signal processing, communications, and image processing.

3. How is Parseval's theorem related to the Fourier transform?

Parseval's theorem is closely related to the Fourier transform, as it provides a way to calculate the energy or power of a signal in the frequency domain using the Fourier transform. It also helps us understand the relationship between a signal and its Fourier transform.

4. What is the proof of Parseval's theorem?

The proof of Parseval's theorem involves using the properties of the Fourier transform, such as linearity and convolution, along with some mathematical manipulation. It can be shown that the integral of the squared magnitude of a signal in the time domain is equal to the integral of the squared magnitude of its Fourier transform in the frequency domain.

5. How is Parseval's theorem used in practice?

Parseval's theorem is used in various applications, such as signal analysis, spectral estimation, and filter design. It allows us to analyze the frequency content of a signal and make decisions based on its energy or power in the frequency domain. It is also used in the development of various signal processing algorithms and techniques.

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