10.2 Bessel's inequality and Parseval's identity

Bessel's inequality and Parseval's identity are key concepts in Hilbert spaces. They connect the Fourier coefficients of a vector to its norm, showing how well a vector can be approximated by its projections onto orthonormal vectors.

These ideas are crucial for understanding orthogonal expansions in Hilbert spaces. They help us measure how much information is captured when we break down a vector into its components along an orthonormal basis.

Bessel's Inequality and Parseval's Identity

Inequality and Identity Relating Fourier Coefficients

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• Bessel's inequality states that for an orthonormal sequence $\{e_n\}$ in an inner product space and any vector $x$, the sum of the squares of the Fourier coefficients is less than or equal to the squared norm of $x$:
  • $\sum_{n=1}^{\infty} |\langle x, e_n \rangle|^2 \leq ||x||^2$
  • Provides an upper bound on the sum of the squares of the Fourier coefficients
• Parseval's identity is a special case of Bessel's inequality when the orthonormal sequence $\{e_n\}$ is complete
  • States that the sum of the squares of the Fourier coefficients is equal to the squared norm of $x$:
    • $\sum_{n=1}^{\infty} |\langle x, e_n \rangle|^2 = ||x||^2$
  • Holds for any orthonormal basis of a Hilbert space
• Fourier coefficients $\langle x, e_n \rangle$ represent the projection of a vector $x$ onto the orthonormal basis vectors $e_n$
  • Measure how much of each basis vector is present in $x$
  • Example: For a square-integrable function $f(t)$ and an orthonormal basis $\{\phi_n(t)\}$, the Fourier coefficients are given by $c_n = \int_{-\infty}^{\infty} f(t) \phi_n^*(t) dt$

Completeness and Orthonormal Bases

• Completeness relation states that if $\{e_n\}$ is an orthonormal basis for a Hilbert space $H$, then any vector $x \in H$ can be expressed as a linear combination of the basis vectors:
  • $x = \sum_{n=1}^{\infty} \langle x, e_n \rangle e_n$
  • Implies that the orthonormal basis spans the entire space
• An orthonormal basis is a set of vectors that are orthogonal (perpendicular) to each other and have unit norm
  • $\langle e_n, e_m \rangle = \delta_{nm}$, where $\delta_{nm}$ is the Kronecker delta (1 if $n=m$, 0 otherwise)
  • Example: The standard basis vectors $\{(1,0,0), (0,1,0), (0,0,1)\}$ form an orthonormal basis for $\mathbb{R}^3$

Signal Energy and Plancherel's Theorem

Energy and Inner Products

• Energy of a signal $x(t)$ is defined as the integral of the squared magnitude of the signal:
  • $E = \int_{-\infty}^{\infty} |x(t)|^2 dt$
  • Represents the total power of the signal over all time
• Plancherel's theorem relates the energy of a signal to its Fourier transform $X(f)$:
  • $\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$
  • States that the energy of a signal in the time domain is equal to the energy in the frequency domain
• Inner product space is a vector space with an inner product operation that satisfies certain properties (conjugate symmetry, linearity, and positive definiteness)
  • Allows for the definition of orthogonality and norm (length) of vectors
  • Example: The space of square-integrable functions $L^2(\mathbb{R})$ with the inner product $\langle f, g \rangle = \int_{-\infty}^{\infty} f(t) g^*(t) dt$ is an inner product space

Orthonormal Bases and Plancherel's Theorem

• Orthonormal bases play a crucial role in the study of signal energy and Plancherel's theorem
  • Provide a way to decompose a signal into its frequency components
  • Allow for the computation of signal energy using Parseval's identity
• Plancherel's theorem can be seen as a consequence of Parseval's identity when applied to the Fourier basis
  • The Fourier basis $\{e^{i2\pi ft}\}_{f \in \mathbb{R}}$ is an orthonormal basis for the space of square-integrable functions
  • Parseval's identity relates the energy of a signal to the sum of the squares of its Fourier coefficients, which leads to Plancherel's theorem