MATH 414 Lecture 27

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Wavelets

Haar scaling function: 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 & x \in [0,1) \\ 0 & \mbox{otherwise} \end{cases}}

Shifts: 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-k)}

"Sampling" spaces

• Integers: 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 \in L^2 ~\mid~ f\ \mbox{is constant on}\ k \le x < k+1, k \in \mathbb{Z} \right\}} (sampling interval has unit length 1)
• Half-Integers: $V_{1}=\left\{f\in L^{2}~\mid ~f\ {\mbox{is constant on}}\ {\frac {k}{2}}\leq x<{\frac {k+1}{2}}\right\}$ (sampling interval has length 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 \frac{1}{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 V_j = \left\{ f \in L^2 ~\mid~ f\ \mbox{is constant on}\ \frac{k}{2^j} \le x < \frac{k+1}{2^j} \right\}} (sampling interval has length 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 2^{-j}} )

Note: As 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 j} increases, the "sampling" occurs at finer scales.

Properties

The following 5 properties institute a multi-resolution analysis.

Definition: The 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_j} 's are called approximation spaces (or scaling spaces)

Nesting

Theorem. [Nesting.] The spaces 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_j} are nested:

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 \dots \subset V_{j-1} \subset V_j \subset V_{j+1} \subset \dots}

Proof by induction. 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 \subset V_1} .

If 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 f \in V} , then 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 f} is constant on 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 2^{-j} \, k \le x < 2^{-j} \, \left( k + 1 \right)} . This implies 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 f} is constant on 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 2^{-(j+1)}\,k \le x < 2^{-(j+1)} \, \left( k+1 \right)} and on 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 2^{-(j+1)} \, \left( k+1 \right) \le x < 2^{-(j+1)} \, \left( k + 2 \right)} . Hence 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_j \subset V_{j+1}}

The hypothesis follows by induction on 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 j} .

quod erat demonstrandum

Density

One can approximate any 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 f} 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} arbitrarily well by functions 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 V_j} if 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 j} is large enough.

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 \bigcup_{j\in \mathbb{Z}} V_j = L^2}

Separation

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 \bigcap_{j \in \mathbb{Z}} V_j = \left\{ 0 \right\}}

Orthonormal Basis

Theorem. The set 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\{ 2^{\frac{j}{2}} \, \phi \left(2^j \, x - k \right) \right\}_{k=-\infty}^\infty} is an orthonormal basis for 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_j}

Proof. We know from last time 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\langle \phi(x-k), \phi(x-\ell) \right\rangle = \delta_{k,\ell}}

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 2^{\frac{j}{2}} \, \phi \left( 2^j \, x - k \right) \, \phi \left( 2^j - \ell \right) \cdot 2^{ \frac{j}{2}} \,\mathrm{d}x = 2^j \, \int_{-\infty}^\infty \phi \left( 2^j \, x - k \right) \, \phi \left( 2^j \, x - \ell \right) \,\mathrm{d}x}

If we let 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 u = 2^j \, x} , then

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 2^j \, \int_{-\infty}^\infty \phi \left( 2^j \, x - k \right) \, \phi \left( 2^j \, x - \ell \right) \,\mathrm{d}x = \int_{-\infty}^\infty \phi \left( u-k \right) \, \phi \left( u - \ell \right) \,\mathrm{d}u = \delta_{k,\ell}}

quod erat demonstrandum

Scaling Property

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 f(x) \in V_j} if and only if 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 f (2^{-j} \, x) \in V_0}

We know if 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 f \in V_j} , then 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 f \in V_{j+1}} (nesting).

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 f = \sum_{k=-\infty}^\infty \left\langle 2^{ \frac{j}{2} } \, \phi \left( 2^j \, \left( \cdot \right) - k \right), f \right\rangle_{L^2} \, 2^{ \frac{j}{2} } \, \phi \left( 2^j \, x - k \right)}

Let 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 a_k^j = 2^{ \frac{j}{2} } \, \left\langle \phi \left( 2^j \, \left( \cdot \right) - k \right), f \right\rangle_{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 \begin{align} a_k^j &= 2^j \int_{-\infty}^\infty \phi \left( 2^j \, x - k \right) \, f(x) \,\mathrm{d}x \\ &= 2^j \, \int_{2^{-j}\,k}^{2^{-j}\,(k+1)} \, f(x) \, \,\mathrm{d}x \\ &= \frac{1}{2^{-j}} \, \int_{2^{-j}\,k}^{2^{-j}\,(k+1)} \, f(x) \, \,\mathrm{d}x \end{align}}

This integral represents the average value over the interval 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[ 2^{-j} \, k, 2^{-j} \, (k+1) \right)}