Wavelets and Applications of Functional Analysis

Motivation: Local Analysis of Signals

Fourier analysis "sees" frequencies globally: a discontinuity at a single point is "smeared" across all coefficients. Wavelets are a "microscope with adjustable magnification": they analyze the signal locally in both time and scale. High frequencies correspond to short wavelets (good time resolution); low frequencies to long ones (good frequency resolution). This enables efficient compression of signals with discontinuities (JPEG 2000, medical imaging).

Limitations of Fourier and the Uncertainty Principle

Heisenberg's Uncertainty Principle: $\sigma_t \cdot \sigma_\omega \geq \frac{1}{4\pi}$, where $\sigma_t, \sigma_\omega$ are the root mean square deviations of $f(t)$ and $\hat{f}(\omega)$. It is impossible to have both good time and good frequency resolution simultaneously.

Windowed Fourier (STFT): $f(t) \cdot g(t-\tau) \to$ Fourier. With a fixed window $g \to$ same resolution for all frequencies. This is suboptimal for signals with variable frequency.

Wavelets: adapt the window size to the scale. High frequencies correspond to a narrow wavelet (precise localization in time). Low frequencies correspond to a wide wavelet (precise localization in frequency).

Wavelet Transform

Mother wavelet $\psi(t)$: $\int \psi,dt = 0$ (zero mean), $|\psi|{L^2}=1$. Admissibility condition: $C\psi = \int \frac{|\hat\psi(\omega)|^2}{|\omega|} d\omega < \infty$.

Continuous wavelet transform: $(Wf)(a,b) = \frac{1}{\sqrt{|a|}} \int f(t) \psi^*\left(\frac{t-b}{a}\right) dt.$

Here $a$ is the scale ($a \to 0$: high frequencies), $b$ is the shift.

Inversion: $f(t) = \frac{1}{C_\psi}\iint (Wf)(a,b)\cdot \frac{1}{\sqrt{|a|}}\psi\left(\frac{t-b}{a}\right)\frac{da,db}{a^2}$.

Haar Wavelet and FWT

Haar wavelet: $\psi(t) = +1$ on $[0,1/2)$, $-1$ on $[1/2,1)$, $0$ otherwise. The simplest; not smooth.

Daubechies wavelets (db2, db4, ...): compact support, $N$ vanishing moments ($\psi \perp$ polynomials of degree

lt; N$), maximal smoothness. Used in JPEG 2000.

FWT (Fast Wavelet Transform): $O(N)$ via filter bank: convolution with $h$ (low-pass) and $g$ (high-pass) + decimation. Faster than FFT.

Numerical Example

Problem: Single-level Haar wavelet transform for $x = (4, 6, 10, 12, 8, 6, 5, 5)$.

Step 1. Haar filters: averaging (low-pass): $cA = (x_{2k-1} + x_{2k})/2$; difference (high-pass): $cD = (x_{2k-1} - x_{2k})/2$.

Step 2. Level 1. Pairs: $(4,6), (10,12), (8,6), (5,5)$.

• $cA_1 = \left[(4+6)/2, (10+12)/2, (8+6)/2, (5+5)/2\right] = [5, 11, 7, 5]$.
• $cD_1 = \left[(4-6)/2, (10-12)/2, (8-6)/2, (5-5)/2\right] = [-1, -1, 1, 0]$.

Step 3. Level 2. Pairs $cA_1$: $(5,11), (7,5)$.

• $cA_2 = \left[(5+11)/2, (7+5)/2\right] = [8, 6]$.
• $cD_2 = \left[(5-11)/2, (7-5)/2\right] = [-3, 1]$.

Step 4. Level 3. Pair $cA_2$: $(8,6)$.

• $cA_3 = [7]$. $cD_3 = [1]$.

Step 5. Full coefficient vector: $[cA_3, cD_3, cD_2, cD_1] = [7, 1, -3, 1, -1, -1, 1, 0]$.

Step 6. Compression: we set $cD_1 = [0,0,0,0]$ (small details). Remaining: $[7, 1, -3, 1, 0,0,0,0]$. Inverse: $cA_2 = [8,6]$, $cA_1 = [(8+3)/2, (8-3)/2, (6-1)/2, (6+1)/2]$? No: $cA_1$ is recovered from $cA_2$ and $cD_2$: $[8-3, 8+3, 6-1, 6+1] = [5,11,5,7]$. Then from $cA_1=[5,11,5,7]$ and $cD_1=[0,0,0,0]$: $x \approx [5,5, 11,11, 5,5, 7,7] =$ average by pairs. Compression error equals discarded $cD_1$.

Step 7. Energy: $|x|^2 = 4^2+6^2+10^2+12^2+8^2+6^2+5^2+5^2 = 16+36+100+144+64+36+25+25 = 446$. $|\text{coefficients}|^2 = 7^2+1^2+3^2+1^2+1^2+1^2+1^2+0^2 = 49+1+9+1+1+1+1 = 63$. Oops! By the Haar–Parseval theorem: $|x|^2 = n \cdot |\text{coefficients}|^2 \to 446/8 = 55.75 \approx 63$? The difference is due to normalization (without $\sqrt{2}$). With proper normalization (divide by $\sqrt{2}$): $|cA_{i}|^2 + |cD_{i}|^2 = |cA_{i-1}|^2 \to 446/8 = 55.75$. Good: discarding $cD_1$ lost: $1+1+1+0=3$ units out of $55.75 \to \sim 5%$ energy. Good compression!

Real-World Application

JPEG 2000 uses Daubechies wavelets for image compression. Unlike JPEG, there are no "block artifacts". Used in digital cinemas (DCI), museum archives (the Louvre), and medical imaging (DICOM). At 10:1 compression, JPEG 2000 quality significantly surpasses JPEG.

Relation to Other Branches of Mathematics

The construction of orthonormal wavelet bases in $L^2(\mathbb{R})$ directly relies on the general theory of unitary representations of groups and harmonic analysis on locally compact groups. The continuous wavelet transform is an example of an irreducible representation of the affine group of the real line; an analogue of decomposition via systems of quasi-regular representations is described in the works of Mackey and Duflo. Discrete wavelets, constructed through multiresolution analyses (MRA), are connected to the theory of Riesz bases and frames (Paley–Wiener and Freidlin–Kadison theorems in modified form).

In differential equations, wavelet decompositions are used in adaptive schemes for solving elliptic and parabolic problems. In the works of A. Cohen and R. DeVore, it has been shown that for operators of the Laplace type and more general operators with elliptic coefficients, the operator matrix in a wavelet basis is nearly sparse; this leads to estimates of convergence of wavelet Galerkin schemes and results on the nonlinear best approximation of solutions. In spectral problems for Schrödinger operators, wavelet bases with controlled support regularity are used, allowing the construction of matrices with rapidly decaying off-diagonal elements.

From a probabilistic point of view, wavelets play a role in the description of fractal processes and fields. For fractional Brownian motion, it has been shown (Meyer, Utakawa) that wavelet coefficients are almost independent at different scales, which allows one to derive Hölder regularity estimates and describe multifractal spectra. In signal statistics, methods of thresholding wavelet coefficients (D. Donoho, I. Johnstone) lead to asymptotically optimal estimates in denoising problems and recovery of functions in Besov-type spaces.

Algebraic aspects arise in multilevel filter banks. The orthogonality and biorthogonality conditions for filters are formulated in terms of the unitarity of matrix symbols and factorization of polynomials with integer coefficients; here, results from the theory of Laurent polynomials and positive-definite trigonometric polynomials are used (Fejér–Riesz lemma). The topological perspective arises on compact manifolds: the construction of wavelets on the sphere, torus, or Riemannian manifolds reduces to the analysis of eigenfunctions of the Laplacian and constructions compatible with the isometry group (works of NarKiewicz, Hammond, Nedelcu).

Historical Note and Development of the Idea

The first discrete system, close in spirit to wavelets, was introduced by A. Haar in 1910 in the Annales de l’École Normale Supérieure in the context of the study of orthonormal systems in $L^2[0,1]$. For a long time, this construction was regarded as an exotic example accompanying the trigonometric basis. In the 1930s–1940s, Norbert Wiener and Jean Vilenkin discussed group representations and local bases, but systematic development began much later.