MATH 414 Lecture 26

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Wavelets

History

Where did they come from?

Historical work by Yves Meyer

Mathematical aspects

• Haar
• Strömberg
• Meyer
• Battle
• Daubechies
• Mallat

Physical aspects (from Geophysics in compressing seismic traces)

• Morlet and Grossmann

Not frequency-based; scale-based instead

Haar Wavelet Analysis

Haar Scaling function ("father wavelet") $\varphi (x)=\{\begin{array}{ll}1& 0\le x<1\\ 0& \text{otherwise}\end{array}$

Linear combinations of step functions gives space of discretized step functions.

Observe that ${\{\varphi (x-k)\}}_{k=0}^{\mathrm{\infty }}$ forms an orthonormal basis in $L^2$:

• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\varphi (x-k{)}^2\mathrm{d}x={\Vert \varphi (x-k)\Vert }^2=1$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\varphi (x-k)\varphi (x-\ell )\mathrm{d}x=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}0\mathrm{d}x=0$

Subspace Definitions

$V_0=\{f(x)\in L^2\mid f(x)\text{is constant on an interval}k\le x<k+1\}$ $V_1=\{f(x)\in L^2\mid f(x)\text{ is constant in}\frac{k}{2}\le x<\frac{k+1}{2}\}$ (forms space of half-integers)