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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ \phi(x-k) \right\}_{k=0}^{\infty}} forms an orthonormal basis in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} :

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{\infty} \phi(x-k)^2 \,\mathrm{d}x = \left\| \phi(x-k) \right\|^2 = 1}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^{\infty} \phi(x-k) \, \phi(x-\ell) \,\mathrm{d}x = \int_{-\infty}^{\infty} 0 \,\mathrm{d}x = 0}

Subspace Definitions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_0 = \left\{ f(x) \in L^2 ~\mid~ f(x)\ \mbox{is constant on an interval}\ k \le x < k+1 \right\}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_1 = \left\{ f(x) \in L^2 ~\mid~ f(x)\ \mbox{ is constant in}\ \frac{k}{2} \le x < \frac{k+1}{2} \right\}} (forms space of half-integers)