🎵Harmonic Analysis Unit 10 – Hilbert Spaces & Orthogonal Expansions

Key Concepts and Definitions

• Hilbert spaces generalize the notion of Euclidean space to infinite-dimensional vector spaces equipped with an inner product
• Elements of a Hilbert space can be functions, sequences, or abstract vectors satisfying certain properties
• Completeness ensures that every Cauchy sequence in the space converges to an element within the space
• Separability implies the existence of a countable dense subset, allowing approximations by finite-dimensional subspaces
  • For example, the space of square-integrable functions $L^2([a,b])$ is separable
• Orthogonality extends the concept of perpendicularity to abstract vector spaces using the inner product
• Orthonormal bases consist of mutually orthogonal unit vectors spanning the entire Hilbert space
  • Fourier basis $\{e^{inx}\}_{n\in\mathbb{Z}}$ forms an orthonormal basis for $L^2([-\pi,\pi])$

Properties of Hilbert Spaces

• Hilbert spaces are vector spaces over the field of real or complex numbers
• The inner product induces a norm, which measures the length or magnitude of vectors
• Parallelogram law holds in Hilbert spaces: $\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2)$
• Polarization identity expresses the inner product in terms of norms: $\langle x,y \rangle = \frac{1}{4}(\|x+y\|^2 - \|x-y\|^2)$
• Hilbert spaces are complete metric spaces with respect to the distance derived from the norm
  • Convergence in norm implies convergence in the metric: $\|x_n - x\| \to 0 \Rightarrow d(x_n,x) \to 0$
• Bounded linear functionals on a Hilbert space can be represented by inner products (Riesz representation theorem)
• Orthogonal projection onto a closed subspace is well-defined and minimizes the distance to the subspace

Inner Products and Norms

• Inner product is a sesquilinear, conjugate-symmetric, positive-definite map $\langle \cdot,\cdot \rangle: H \times H \to \mathbb{F}$
  • Sesquilinearity: $\langle \alpha x + \beta y, z \rangle = \alpha \langle x,z \rangle + \beta \langle y,z \rangle$
  • Conjugate-symmetry: $\langle x,y \rangle = \overline{\langle y,x \rangle}$
  • Positive-definiteness: $\langle x,x \rangle \geq 0$ with equality iff $x=0$
• Norm induced by the inner product: $\|x\| = \sqrt{\langle x,x \rangle}$
• Cauchy-Schwarz inequality: $|\langle x,y \rangle| \leq \|x\| \|y\|$
  • Equality holds iff $x$ and $y$ are linearly dependent
• Triangle inequality for norms: $\|x+y\| \leq \|x\| + \|y\|$
• Parallelogram law relates norms to inner products: $\|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2)$

Orthogonality and Orthonormal Bases

• Vectors $x$ and $y$ are orthogonal if their inner product is zero: $\langle x,y \rangle = 0$
• Orthogonal complement of a subset $S$ is the set of all vectors orthogonal to every vector in $S$: $S^\perp = \{x \in H: \langle x,y \rangle = 0 \text{ for all } y \in S\}$
• Orthogonal projection of $x$ onto a closed subspace $M$ is the unique vector $P_M(x) \in M$ such that $x - P_M(x) \in M^\perp$
• Orthonormal basis is a basis consisting of orthogonal unit vectors: $\langle e_i,e_j \rangle = \delta_{ij}$
  • Fourier basis $\{e^{inx}\}_{n\in\mathbb{Z}}$ is an orthonormal basis for $L^2([-\pi,\pi])$
• Parseval's identity relates the norm of a vector to its coefficients in an orthonormal basis: $\|x\|^2 = \sum_{n=1}^\infty |\langle x,e_n \rangle|^2$
• Bessel's inequality provides an upper bound for the sum of squared coefficients: $\sum_{n=1}^\infty |\langle x,e_n \rangle|^2 \leq \|x\|^2$

Fourier Series and Expansions

• Fourier series represents a periodic function as an infinite sum of trigonometric functions
  • For $f \in L^2([-\pi,\pi])$, the Fourier series is given by $f(x) = \sum_{n=-\infty}^\infty c_n e^{inx}$
• Fourier coefficients are the inner products of the function with the Fourier basis: $c_n = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) e^{-inx} dx$
• Dirichlet kernel $D_N(x) = \sum_{n=-N}^N e^{inx}$ plays a role in the convergence of Fourier series
• Fejér kernel $F_N(x) = \frac{1}{N+1} \sum_{k=0}^N D_k(x)$ provides a smoother approximation to the function
• Parseval's theorem for Fourier series: $\|f\|^2 = \sum_{n=-\infty}^\infty |c_n|^2$
• Fourier transform extends the concept of Fourier series to non-periodic functions on the real line
  • For $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$, the Fourier transform is $\hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i x \xi} dx$

Convergence in Hilbert Spaces

• Convergence in norm: a sequence $\{x_n\}$ converges to $x$ in norm if $\lim_{n\to\infty} \|x_n - x\| = 0$
• Weak convergence: $\{x_n\}$ converges weakly to $x$ if $\lim_{n\to\infty} \langle x_n,y \rangle = \langle x,y \rangle$ for all $y \in H$
  • Weak convergence is denoted by $x_n \rightharpoonup x$
• Strong convergence (convergence in norm) implies weak convergence, but the converse is not always true
• Riesz-Fischer theorem states that $L^2([a,b])$ is complete, ensuring the convergence of Cauchy sequences
• Bessel's inequality and Parseval's identity provide criteria for convergence of Fourier series and expansions
  • If $\sum_{n=1}^\infty |\langle f,e_n \rangle|^2 < \infty$, then the Fourier series of $f$ converges in $L^2$-norm
• Fejér's theorem: the Fejér means of a continuous function converge uniformly to the function

Applications in Harmonic Analysis

• Fourier analysis decomposes functions into their frequency components, enabling the study of signals and systems
• Hilbert spaces provide a framework for studying partial differential equations (PDEs) and boundary value problems
  • Eigenfunctions of differential operators often form orthonormal bases for function spaces
• Wavelet analysis uses localized basis functions to represent signals at different scales and positions
  • Wavelet bases are orthonormal bases for $L^2(\mathbb{R})$ with good time-frequency localization properties
• Spectral theory investigates the properties of linear operators on Hilbert spaces
  • Spectrum of an operator generalizes the concept of eigenvalues to infinite-dimensional spaces
• Quantum mechanics formulates states as vectors in a Hilbert space and observables as self-adjoint operators
  • Inner product encodes the probability amplitudes, and eigenvalues correspond to possible measurement outcomes

Common Pitfalls and Tips

• Ensure that the inner product satisfies the required properties (sesquilinearity, conjugate-symmetry, positive-definiteness)
• Be cautious when interchanging limits and infinite sums or integrals, as convergence issues may arise
• Weak convergence does not imply strong convergence; additional conditions may be needed
• Orthogonality does not imply linear independence in infinite-dimensional spaces
  • For example, the functions $\{x^n\}_{n=0}^\infty$ are orthogonal but not linearly independent in $L^2([0,1])$
• Verify that a given set of vectors forms an orthonormal basis by checking orthogonality, normalization, and completeness
• Pay attention to the underlying field (real or complex) when working with inner products and norms
• Fourier series of a continuous function may not converge pointwise at every point; consider other notions of convergence (e.g., $L^2$-norm, uniform)