11.1 Plancherel theorem for Fourier transforms

The Plancherel theorem is a game-changer for Fourier transforms. It shows that these transforms keep the inner product and norm of functions intact, letting us switch between time and frequency domains without losing info.

This theorem proves the Fourier transform is an isometry on L^2 functions. It means we can analyze functions in the frequency domain while keeping their original properties, which is super useful in signal processing and other fields.

Plancherel Theorem and Fourier Transform

Plancherel Theorem and its Significance

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• States that the Fourier transform is an isometry between the Hilbert space $L^2(\mathbb{R})$ and itself
• Implies that the Fourier transform preserves the inner product and norm of functions in $L^2(\mathbb{R})$
• Establishes a fundamental connection between a function and its Fourier transform
• Allows for the analysis of functions in the frequency domain while preserving their properties in the original domain

Fourier Transform and $L^2$ Functions

• The Fourier transform maps a function $f(x)$ from the time/spatial domain to the frequency domain, denoted as $\hat{f}(\xi)$
• Defined as $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i x \xi} dx$
• $L^2$ functions are square-integrable functions, meaning $\int_{-\infty}^{\infty} |f(x)|^2 dx < \infty$
• The space of $L^2$ functions, denoted as $L^2(\mathbb{R})$, forms a Hilbert space with the inner product $\langle f, g \rangle = \int_{-\infty}^{\infty} f(x)\overline{g(x)} dx$
• The Fourier transform is well-defined for $L^2$ functions and maps them to another $L^2$ function

Isometry and its Implications

• An isometry is a mapping that preserves distances between points in a metric space
• In the context of the Plancherel theorem, the Fourier transform is an isometry on $L^2(\mathbb{R})$
• Isometry implies that $\|\hat{f}\|_{L^2} = \|f\|_{L^2}$, where $\|\cdot\|_{L^2}$ denotes the $L^2$ norm
• Preserves the inner product: $\langle \hat{f}, \hat{g} \rangle = \langle f, g \rangle$ for any $f, g \in L^2(\mathbb{R})$
• Allows for the analysis of functions in the frequency domain without losing information about their properties in the original domain

Hilbert Space and Norm Preservation

Hilbert Space and its Properties

• A Hilbert space is a complete inner product space, which is a vector space equipped with an inner product that induces a norm
• $L^2(\mathbb{R})$ is an example of a Hilbert space, where the inner product is defined as $\langle f, g \rangle = \int_{-\infty}^{\infty} f(x)\overline{g(x)} dx$
• Hilbert spaces have desirable properties, such as the existence of orthonormal bases and the ability to define projections onto closed subspaces
• The completeness property ensures that Cauchy sequences converge to a limit within the space

Norm Preservation and its Significance

• The Plancherel theorem states that the Fourier transform preserves the $L^2$ norm of functions
• Norm preservation means that $\|\hat{f}\|_{L^2} = \|f\|_{L^2}$ for any $f \in L^2(\mathbb{R})$
• Implies that the energy of a function is the same in both the time/spatial domain and the frequency domain
• Allows for the analysis of functions in the frequency domain without losing information about their energy or magnitude
• Enables the use of Parseval's identity, which relates the inner product of functions to the inner product of their Fourier transforms

Spectral Density and its Relation to the Fourier Transform

• The spectral density, also known as the power spectral density, describes the distribution of power or energy across different frequencies
• For a function $f \in L^2(\mathbb{R})$, the spectral density is given by $|\hat{f}(\xi)|^2$, where $\hat{f}$ is the Fourier transform of $f$
• The Plancherel theorem implies that the total energy of a function can be computed using its spectral density: $\int_{-\infty}^{\infty} |f(x)|^2 dx = \int_{-\infty}^{\infty} |\hat{f}(\xi)|^2 d\xi$
• Spectral density provides valuable information about the frequency content and energy distribution of a function
• Useful in various applications, such as signal processing, where the frequency components of a signal are of interest