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") 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 \phi(x) = \begin{cases} 1 & 0 \le 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)