🧐Functional Analysis Unit 14 Review

Wavelets and frames are powerful tools for analyzing signals and functions in Hilbert spaces. They provide flexible ways to decompose and represent complex data, offering unique insights into signal characteristics at different scales and positions.

These techniques have revolutionized signal processing, image compression, and numerical analysis. By exploiting the sparsity and multi-resolution nature of wavelet representations, we can efficiently solve problems in various fields of mathematics and engineering.

Wavelets and Frames in Hilbert Spaces

Wavelets and frames in Hilbert spaces

Wavelets are functions $\psi \in L^2(\mathbb{R})$ $ψ \in L^{2} (R)$ used to decompose and analyze signals or functions
- Obtained by translating and dilating a mother wavelet $\psi$ $ψ$
  - Translation shifts the wavelet: $\psi_{a,b}(x) = \psi(x-b)$
  - Dilation scales the wavelet: $\psi_{a,b}(x) = \frac{1}{\sqrt{|a|}}\psi(\frac{x-b}{a})$ , where $a,b \in \mathbb{R}, a \neq 0$
- Must satisfy admissibility condition: $\int_{-\infty}^{\infty} \frac{|\hat{\psi}(\omega)|^2}{|\omega|} d\omega < \infty$ $\int_{- \infty}^{\infty} \frac{∣ ψ ^ ( ω ) ∣ ^{2}}{∣ ω ∣} d ω < \infty$ , where $\hat{\psi}$ $\hat{ψ}$ is the Fourier transform of $\psi$ $ψ$
  - Ensures wavelet can be used for signal reconstruction
Frames are a sequence $\{f_n\}_{n=1}^{\infty}$ ${f_{n}}_{n = 1}^{\infty}$ in a Hilbert space $H$ $H$ that provide stable, redundant representations
- Exist constants $A, B > 0$ $A, B > 0$ such that for all $f \in H$ $f \in H$ : $A\|f\|^2 \leq \sum_{n=1}^{\infty} |\langle f, f_n \rangle|^2 \leq B\|f\|^2$ $A ∥ f ∥^{2} \leq \sum_{n = 1}^{\infty} ∣ ⟨ f, f_{n} ⟩ ∣^{2} \leq B ∥ f ∥^{2}$
  - $A$ and $B$ are frame bounds that quantify stability and redundancy
- Tight frames have equal frame bounds $A = B$
- Parseval frames are tight frames with $A = B = 1$ , behave like orthonormal bases

Construction of wavelets and frames

Haar wavelet is a simple example of an orthonormal wavelet basis for $L^2(\mathbb{R})$ $L^{2} (R)$
- Mother wavelet: $\psi(x) = \begin{cases} 1, & 0 \leq x < \frac{1}{2} \\ -1, & \frac{1}{2} \leq x < 1 \\ 0, & \text{otherwise} \end{cases}$
- Haar wavelets: $\psi_{j,k}(x) = 2^{j/2}\psi(2^jx-k)$ $ψ_{j, k} (x) = 2^{j /2} ψ (2^{j} x - k)$ , where $j,k \in \mathbb{Z}$ $j, k \in Z$
  - $j$ controls scale (dilation) and $k$ controls position (translation)
Gabor frames are constructed from a window function $g \in L^2(\mathbb{R})$ $g \in L^{2} (R)$ and lattice parameters $a,b > 0$ $a, b > 0$
- Gabor atoms: $g_{m,n}(x) = e^{2\pi ibnx}g(x-am)$ $g_{m, n} (x) = e^{2 πibn x} g (x - am)$ , where $m,n \in \mathbb{Z}$ $m, n \in Z$
  - $m$ and $n$ control time and frequency shifts, respectively
- The set $\{g_{m,n}\}_{m,n \in \mathbb{Z}}$ ${g_{m, n}}_{m, n \in Z}$ forms a frame for $L^2(\mathbb{R})$ $L^{2} (R)$ if $ab < 1$ $ab < 1$
  - Oversampling in time-frequency plane ensures completeness and stability

Wavelets and frames in Hilbert spaces, Frontiers | Exploring the Abnormal Modulation of the Autonomic Systems during Nasal Flow ...

Convergence of wavelet expansions

For $f \in L^2(\mathbb{R})$ $f \in L^{2} (R)$ , the wavelet expansion is given by $f = \sum_{j,k \in \mathbb{Z}} \langle f, \psi_{j,k} \rangle \psi_{j,k}$ $f = \sum_{j, k \in Z} ⟨ f, ψ_{j, k} ⟩ ψ_{j, k}$
- Wavelet coefficients $\langle f, \psi_{j,k} \rangle$ capture signal information at different scales and positions
Convergence in $L^2(\mathbb{R})$ $L^{2} (R)$ norm: $\lim_{N \to \infty} \|\sum_{j,k \in \mathbb{Z}, |j|,|k| \leq N} \langle f, \psi_{j,k} \rangle \psi_{j,k} - f\| = 0$ $lim_{N \to \infty} ∥ \sum_{j, k \in Z, ∣ j ∣, ∣ k ∣ \leq N} ⟨ f, ψ_{j, k} ⟩ ψ_{j, k} - f ∥ = 0$
- Partial sums converge to the original signal as more terms are included
Wavelet coefficients are stable, bounded by the $L^2$ norm of the signal: $|\langle f, \psi_{j,k} \rangle| \leq C\|f\|$
Frame expansions converge in Hilbert space norm: $\lim_{N \to \infty} \|\sum_{n=1}^{N} \langle f, S^{-1}f_n \rangle f_n - f\| = 0$ $lim_{N \to \infty} ∥ \sum_{n = 1}^{N} ⟨ f, S^{- 1} f_{n} ⟩ f_{n} - f ∥ = 0$
- $S$ is the frame operator, $S^{-1}$ is its inverse
- Frame coefficients $\langle f, S^{-1}f_n \rangle$ are stable, bounded by the Hilbert space norm of $f$

Applications of wavelets and frames

Signal processing uses wavelet transforms for:
1. Denoising: Thresholding wavelet coefficients to remove noise
2. Compression: Exploiting sparsity of wavelet coefficients to reduce data size
3. Feature extraction: Identifying important signal characteristics at different scales
Image compression algorithms (JPEG2000) use wavelets to:
1. Decompose image into wavelet coefficients
2. Threshold and quantize coefficients to achieve high compression ratios
3. Reconstruct image with minimal perceptual loss
Numerical analysis employs wavelets and frames for:
- Adaptive mesh refinement in finite element methods
  - Wavelets identify regions requiring higher resolution
- Efficient representation and computation of operators and functions
  - Wavelet bases can diagonalize certain operators
- Discretization and solution of partial differential equations
  - Frames provide stable, flexible discretizations of function spaces

🧐Functional Analysis Unit 14 Review