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") $\phi (x)={\begin{cases}1&0\leq x<1\\0&{\mbox{otherwise}}\end{cases}}$

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

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

• $\int _{-\infty }^{\infty }\phi (x-k)^{2}\,\mathrm {d} x=\left\|\phi (x-k)\right\|^{2}=1$
• $\int _{-\infty }^{\infty }\phi (x-k)\,\phi (x-\ell )\,\mathrm {d} x=\int _{-\infty }^{\infty }0\,\mathrm {d} x=0$

Subspace Definitions

$V_{0}=\left\{f(x)\in L^{2}~\mid ~f(x)\ {\mbox{is constant on an interval}}\ k\leq x<k+1\right\}$ $V_{1}=\left\{f(x)\in L^{2}~\mid ~f(x)\ {\mbox{ is constant in}}\ {\frac {k}{2}}\leq x<{\frac {k+1}{2}}\right\}$ (forms space of half-integers)