🎵Harmonic Analysis Unit 11 – Plancherel Theorem and Parseval's Identity

Key Concepts and Definitions

• Fourier transform converts a function from the time domain to the frequency domain, representing it as a sum of sinusoidal components
• Inverse Fourier transform converts a function from the frequency domain back to the time domain
• $L^2$ space consists of square-integrable functions, where the integral of the square of the absolute value of the function is finite
  • Functions in $L^2$ have finite energy and are important in signal processing and quantum mechanics
• Unitary operator is a linear transformation that preserves inner products and norms, satisfying $U^*U = UU^* = I$
• Isometry is a distance-preserving transformation between metric spaces, preserving the length of curves and the distance between points
• Orthonormal basis is a set of vectors that are orthogonal (perpendicular) to each other and have unit length (normalized)
  • Orthonormal bases provide a convenient way to represent vectors and functions in a space

Historical Background

• Joseph Fourier introduced the concept of representing functions as sums of sinusoids in the early 19th century while studying heat transfer
• Plancherel's theorem, named after Michel Plancherel, was first published in 1910 as part of his work on the Fourier transform and its properties
• Parseval's identity, named after Marc-Antoine Parseval, is a special case of Plancherel's theorem for orthonormal bases
  • Parseval's identity was originally discovered in the late 18th century in the context of Fourier series
• The development of Plancherel's theorem and Parseval's identity was influenced by the work of mathematicians such as Hilbert, Riesz, and Fischer
• These results have played a crucial role in the development of harmonic analysis, signal processing, and quantum mechanics throughout the 20th and 21st centuries

Statement of Plancherel Theorem

• Plancherel's theorem states that the Fourier transform is an isometry between the $L^2$ spaces of the time and frequency domains
• Formally, for any function $f \in L^2(\mathbb{R})$, the following equality holds: $\int_{-\infty}^{\infty} |f(x)|^2 dx = \int_{-\infty}^{\infty} |\hat{f}(\xi)|^2 d\xi$
  • Here, $\hat{f}$ denotes the Fourier transform of $f$
• The theorem implies that the Fourier transform preserves the inner product and norm of functions in $L^2$
• Plancherel's theorem can be generalized to other function spaces and transforms, such as the Fourier series and the discrete Fourier transform
• The theorem provides a powerful tool for analyzing the energy distribution of signals in the frequency domain

Parseval's Identity Explained

• Parseval's identity is a special case of Plancherel's theorem for orthonormal bases in a Hilbert space
• For an orthonormal basis $\{e_n\}_{n=1}^{\infty}$ in a Hilbert space $H$ and any $x \in H$, Parseval's identity states: $\sum_{n=1}^{\infty} |\langle x, e_n \rangle|^2 = \|x\|^2$
  • Here, $\langle \cdot, \cdot \rangle$ denotes the inner product, and $\|\cdot\|$ is the norm induced by the inner product
• In the context of Fourier series, Parseval's identity relates the sum of the squares of the Fourier coefficients to the integral of the square of the function
• Parseval's identity is essential in proving the completeness and orthonormality of various function systems, such as trigonometric functions and wavelets
• The identity has applications in signal processing, where it is used to calculate the energy of a signal in terms of its Fourier coefficients

Proof Techniques and Approaches

• The proof of Plancherel's theorem typically involves showing that the Fourier transform is a unitary operator on $L^2$
• One approach is to use the density of compactly supported smooth functions in $L^2$ and prove the theorem for this dense subset
  • The result can then be extended to all of $L^2$ by a continuity argument
• Another approach is to use the spectral theorem for unbounded self-adjoint operators, considering the Fourier transform as a multiplication operator in the frequency domain
• Parseval's identity can be proved using the orthonormality of the basis and the properties of the inner product
  • For Fourier series, the proof often involves integrating the square of the series term by term and using the orthogonality of the trigonometric functions
• Proofs of Plancherel's theorem and Parseval's identity often rely on techniques from functional analysis, such as the Riesz representation theorem and the Lebesgue dominated convergence theorem

Applications in Signal Processing

• Plancherel's theorem and Parseval's identity are fundamental in signal processing, allowing for the analysis of signals in both time and frequency domains
• The theorems provide a way to calculate the energy of a signal using its Fourier transform or Fourier series coefficients
  • This is useful in determining the power spectrum and energy distribution of a signal
• In image processing, Parseval's identity is used to justify the use of the discrete Fourier transform (DFT) for image compression and filtering
• The theorems are also applied in the design of digital filters, as they relate the energy of the input and output signals
• In wireless communications, Parseval's identity is used to analyze the performance of orthogonal frequency-division multiplexing (OFDM) systems
  • OFDM is a method for transmitting digital data using orthogonal subcarriers, and Parseval's identity ensures the preservation of signal energy

Connections to Other Theorems

• Plancherel's theorem and Parseval's identity are closely related to the Pythagorean theorem in Hilbert spaces
  • The Pythagorean theorem states that for orthogonal vectors, the square of the norm of their sum equals the sum of the squares of their norms
• The theorems are also connected to the spectral theorem for self-adjoint operators, which provides a way to represent operators using eigenvalues and eigenvectors
• The Riesz-Fischer theorem, which states that the space of square-summable sequences is isomorphic to $L^2$, is another related result
• Plancherel's theorem can be seen as a generalization of the Parseval's identity for the Fourier transform, extending the result from orthonormal bases to the continuous Fourier transform
• The theorems are also related to the uncertainty principle in signal processing and quantum mechanics, which limits the simultaneous localization of a function and its Fourier transform

Practice Problems and Examples

• Prove Parseval's identity for the Fourier series of a periodic function $f(x)$ with period $2\pi$
• Verify Plancherel's theorem for the Gaussian function $f(x) = e^{-x^2}$, using the fact that its Fourier transform is also a Gaussian
• Show that the energy of a signal $x(t)$ can be calculated using both time and frequency domain integrals: $E = \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$
• Apply Parseval's identity to the Haar wavelet basis, and discuss its implications for wavelet-based signal compression
• Prove that the Fourier basis functions $\{e^{inx}\}_{n=-\infty}^{\infty}$ form an orthonormal basis for $L^2([-\pi, \pi])$, and use Parseval's identity to derive the Fourier series coefficients
• Use Plancherel's theorem to derive the convolution theorem, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms