Wavelet Theory — Mathematics Knowledge System

Overview

Wavelet theory provides a framework for decomposing functions simultaneously in both time (or space) and frequency, overcoming a fundamental limitation of the classical Fourier transform. A wavelet basis {ψⱼ,ₖ} consists of dilates and translates of a single function ψ (the 'mother wavelet'), and functions in L²(ℝ) can be expanded in these bases with sparse, local coefficients. Applications include image compression (JPEG 2000), signal denoising, numerical solutions of PDEs, and quantum mechanics. Ingrid Daubechies constructed the first compactly supported orthogonal wavelets in 1988.

Intuition

The Fourier transform decomposes a signal into pure tones of definite frequency but with no time localisation: a brief click and a sustained note of the same pitch look different in the time domain but the Fourier transform of the click is spread across all frequencies. Wavelets solve this: they are little oscillating waveforms that are localised in both time and frequency. By dilating (stretching) a wavelet one gets lower frequencies; by translating it one 'scans' the time axis. This is like a musical score that records both pitch and time, versus a spectrum that only records pitch.

Formal Definition

Definition

A multiresolution analysis (MRA) of L²(ℝ) is a nested sequence of closed subspaces Vⱼ ⊂ Vⱼ₊₁ generated by an integer-translate-invariant scaling function φ. The wavelet ψ generates the orthogonal complement Wⱼ = Vⱼ₊₁ ⊖ Vⱼ.

\cdots \subset V_{-1} \subset V_0 \subset V_1 \subset \cdots,\quad \overline{\bigcup_j V_j} = L^2(\mathbb{R}),\quad \bigcap_j V_j = \{0\}

Multiresolution analysis (MRA) axioms

\{\phi(\cdot - k)\}_{k \in \mathbb{Z}} \text{ is an orthonormal basis for } V_0

Scaling function generates V₀

\psi_{j,k}(x) = 2^{j/2}\psi(2^j x - k), \quad j, k \in \mathbb{Z}

Wavelet basis functions (dilates and translates)

f = \sum_{j,k \in \mathbb{Z}} \langle f, \psi_{j,k}\rangle\, \psi_{j,k} \quad \text{in } L^2(\mathbb{R})

Wavelet expansion of f ∈ L²(ℝ)

\hat{\psi}(\xi)| \leq C\,(1 + |\xi|)^{-N},\quad \int \psi(x)x^k\,dx = 0 \text{ for } k = 0,\ldots, N-1

N vanishing moments (controls frequency localisation)

Notation

Notation	Meaning
$\psi$	Mother wavelet
$\phi$	Scaling function (father wavelet)
$\psi_{j,k}(x) = 2^{j/2}\psi(2^jx-k)$	Wavelet at scale j and translation k
$V_j$	Approximation space at resolution level j
$W_j = V_{j+1} \ominus V_j$	Detail space at level j

Properties

Parseval's identity for wavelets

\|f\|_{L^2}^2 = \sum_{j,k} |\langle f, \psi_{j,k}\rangle|^2

Vanishing moments and smoothness

\text{If } \psi \text{ has } N \text{ vanishing moments, then } |\langle f, \psi_{j,k}\rangle| = O(2^{-j(N+r)}) \text{ near smooth regions}

Two-scale (refinement) equation

\phi(x) = \sqrt{2}\sum_{k} h_k\, \phi(2x-k), \quad \psi(x) = \sqrt{2}\sum_{k} g_k\, \phi(2x-k)

Worked Examples

1
The Haar scaling function is φ = 1_{[0,1)} and the Haar wavelet is ψ = 1_{[0,1/2)} - 1_{[1/2,1)}. At the first level, the approximation coefficients are averages and detail coefficients are differences.
$a_k = \frac{f_{2k} + f_{2k+1}}{\sqrt{2}},\quad d_k = \frac{f_{2k} - f_{2k+1}}{\sqrt{2}}$
2
Apply to f = [4, 2, 5, 1]:
$a_0 = \frac{4+2}{\sqrt{2}} = 3\sqrt{2},\quad a_1 = \frac{5+1}{\sqrt{2}} = 3\sqrt{2}$
3
Detail coefficients:
$d_0 = \frac{4-2}{\sqrt{2}} = \sqrt{2},\quad d_1 = \frac{5-1}{\sqrt{2}} = 2\sqrt{2}$

✓ Answer

Approximation [3√2, 3√2]; detail coefficients [√2, 2√2]. Note ‖f‖² = 4²+2²+5²+1² = 46 = (3√2)²+(3√2)²+(√2)²+(2√2)² ✓.

Practice Problems

Mediumfree response

State the multiresolution analysis axioms and explain how a wavelet ψ is constructed from a scaling function φ.

Hardfree response

Explain the trade-off between time localisation and frequency localisation in wavelet vs Fourier analysis, and how the Heisenberg uncertainty principle applies.

Common Mistakes

Common Mistake

Thinking wavelets replace the Fourier transform for all purposes

Wavelets are optimal for functions with isolated singularities or transient features. For periodic, globally smooth functions (e.g. solutions of constant-coefficient PDEs), Fourier series remain optimal. The two tools are complementary.

Common Mistake

Assuming any oscillating function is a valid wavelet

A valid (admissible) wavelet must satisfy the admissibility condition: ∫|ψ̂(ξ)|²/|ξ| dξ < ∞, which requires ψ̂(0) = 0, i.e. ψ must have at least one vanishing moment (∫ψ dx = 0). Without this, the continuous wavelet transform cannot be inverted.

Quiz

The key advantage of wavelets over the Fourier transform for analysing signals with transients is:

Wavelets converge fasterWavelets provide simultaneous time and frequency localisationWavelets preserve phase informationWavelets are easier to compute

In a multiresolution analysis, the wavelet basis functions ψⱼ,ₖ are:

Eigenfunctions of the LaplacianDilates and translates of a single mother wavelet ψOrthogonal polynomialsSpherical harmonics

Historical Background

The idea of localised oscillations appeared in Haar's 1910 orthonormal system. The modern theory was developed independently by Morlet (geophysics, 1982), Grossman and Morlet (continuous wavelet transform, 1984), and Meyer (orthonormal wavelets avoiding compactness, 1985). Mallat's multiresolution analysis framework (1989) systematised the construction. Daubechies' 1988 paper 'Orthonormal bases of compactly supported wavelets' provided the compactly supported Daubechies wavelets that dominate applications.

1910
Haar constructs the first wavelet-like orthonormal system
Alfred Haar
1984
Grossmann and Morlet introduce the continuous wavelet transform
Alex Grossmann, Jean Morlet
1985
Meyer constructs the first smooth orthonormal wavelet basis
Yves Meyer
1988
Daubechies constructs compactly supported orthonormal wavelets
Ingrid Daubechies
1989
Mallat introduces multiresolution analysis as a unifying framework
Stéphane Mallat

Summary

Wavelets are dilated and translated copies of a mother wavelet ψ; they provide simultaneous time-frequency localisation.
A multiresolution analysis (MRA) gives a systematic framework: Vⱼ ⊂ Vⱼ₊₁ with detail spaces Wⱼ = Vⱼ₊₁ ⊖ Vⱼ spanned by ψⱼ,ₖ.
Daubechies wavelets are compactly supported, orthonormal, and have N vanishing moments; they dominate applications.
The fast wavelet transform runs in O(N) operations (vs O(N log N) for FFT) using the two-scale filter bank.
N vanishing moments ensure wavelets give sparse representations of smooth functions, enabling compression.

References

BookDaubechies, I. — Ten Lectures on Wavelets (1992), SIAM, CBMS-NSF Regional Conference Series
BookMallat, S. — A Wavelet Tour of Signal Processing, 3rd ed. (2009), Academic Press
BookWojtaszczyk, P. — A Mathematical Introduction to Wavelets (1997), Cambridge University Press
WebsiteWikipedia — Wavelet
WebsiteMathWorld — Wavelet