MATH 414 Lecture 34

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Begin Exam 3 content

Properties of 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 p_k} 's

Assume real-valued; if not, we'll need 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 \overline{p}_k} 's

Scaling function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi (x)=\sum _{k=-\infty }^{\infty }p_{k}\,\phi (2x-k)} , where

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 p_k = 2 \int_{-\infty}^\infty \phi(x) \, \phi(2x-k) \,\mathrm{d}x}

Theorem 5.9:

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 \sum_{k} p_{k-2\ell} p_{k-2m} = 2\delta_{\ell,m}}
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 \sum_{k} p_{k-2\ell}^2 = \sum_{k'} p_{k'}^2 = 2}

Theorem. 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 \sum_{k=-\infty}^\infty p_k = 2}

Proof. Step 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 \begin{align} \int_{-\infty}^\infty \phi(x) \,\mathrm{d}x &= \int_{-\infty}^\infty \sum_{k=-\infty}^\infty p_k \phi(2x-k) \,\mathrm{d}x \\ &= \sum_{k=-\infty}^\infty p_k \, \int_{-\infty}^\infty \phi(2x-k) \,\mathrm{d}x \end{align}}

Step 2: change variables 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 = 2x-k} :

$\int _{-\infty }^{\infty }\phi (2x-k)={\frac {1}{2}}\int _{-\infty }^{\infty }\phi (u)\,\mathrm {d} u={\frac {M}{2}}$

Step 3: It takes work to show this, but we're left with

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 M = \frac{1}{2} \, \sum_{k} p_k \cdot M} , where $M\neq 0$, so 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 1 = \frac{1}{2} \, \sum_{k} p_k} , and 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 \sum_{k} p_k = 2}

quod erat demonstrandum

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 \begin{align} \sum_{\mbox{even}\ k} p_k &= 1 = \sum_{\ell = -\infty}^\infty p_{2\ell} \\ \sum_{\mbox{odd}\ k} p_k &= 1 = \sum_{\ell = -\infty}^\infty p_{2\ell+1} \end{align}}

Example: Haar

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) = \phi(2x) + \phi(2x+1)} , so 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 p_0 = 1, p_1 = 1} .

3. 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 p_0 + p_1 = 2}

4. 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 p_0 = 1} (even), $p_{1}=1$ (odd)

Example: Shannon

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) = \mathrm{sinc}(x) = \begin{cases} \frac{\sin{\pi\,x}}{\pi \, x} & x \ne 0 \\ 1 & x = 0 \end{cases}}

Scaling: 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) = \phi(2x) + \sum_{k=-\infty}^{\infty} \frac{2(-1)^k}{(2k+1) \, \pi} \, \phi(2x-1-2k)}

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 p_k = \begin{cases} 1 & k = 0 \\ \frac{2(-1)^{\frac{\ell-1}{2}}}{\pi\,k} & \mbox{odd}\ \ell = 2k+1 \\ 0 & \mbox{otherwise} \end{cases}}

Now 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 p_0 = 1} is the only even term.

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 f(t) = \sum_{k\in \mathbb{Z}} \frac{-2i}{(2k+1)\pi} \, \mathrm{e}^{(2k+1)\,i\,t}} be the fourier series for the alternating function

Then $f\left({\frac {\pi }{2}}\right)=1=\sum _{k\in \mathbb {Z} }{\frac {-2i}{(2k+1)\pi }}\,\mathrm {e} ^{i\,k\,\pi }\cdot \mathrm {e} ^{i\,{\frac {\pi }{2}}}=\sum _{k\in \mathbb {Z} }{\frac {2(-1)^{k}}{(2k+1)\pi }}$

Daubechies 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 p=2} MRA

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) = \sum_{k=0}^3 p_k \, \phi(2x-k) = p_0 \, \phi(2x) + p_1 \, \phi(2x-1) + p_2 \, \phi(2x-2) + p_3 \, \phi(2x-3)} , where

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 p_k = \left\{ \frac{1 + \sqrt{3}}{4}, \frac{3 + \sqrt{3}}{4}, \frac{3-\sqrt{3}}{4}, \frac{1 - \sqrt{3}}{4} \right\}}

These were derived from the properties, and not from a given 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} .

Wavelets

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 W_j = \left\{ w \in V_{j+1} ~\mid~ w \perp V_j \right\}}

Looking 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 \psi(x)} such that $\left\{\psi (x-k)\right\}_{k\in \mathbb {Z} }$ 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 W_0} .

From this we get 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} } \, \psi \left( 2^{ \frac{j}{2} } \, x - k \right) \right\}_{k \in \mathbb{Z}}} 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 W_j} .

find 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 q_k} 's such 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 \psi(x) = \sum_{k \in \mathbb{Z}} q_k \, \phi(2x-k)} . We want

1. $\int \psi (x-k)\,\phi (x-\ell )\,\mathrm {d} x=0$ for all 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 k,\ell}
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 \psi(x-k) \, \psi(x-\ell) \,\mathrm{d}x = \delta_{k,\ell}}

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 k = \ell} , we get $\int \psi (x)\,\phi (x)\,\mathrm {d} x=0$.

We have

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} \psi(x) &= \sum_{k \in \mathbb{Z}} q_k \, \phi(2x-k) \\ \phi(x) &= \sum_{\ell \in \mathbb{Z}} p_{\ell} \, \phi(2x-\ell) \end{align}}

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 \begin{align} \int_{-\infty}^\infty \psi(x) \, \phi(x) \,\mathrm{d}x &= \int_{-\infty}^\infty \sum_k q_k \, \phi(2x-k) \, \phi(x) \,\mathrm{d}x \\ 0 &= \sum_{k} \frac{1}{2} \, q_k \underbrace{ \left( 2 \int_{-\infty}^\infty \phi(x) \phi(2x-k) \,\mathrm{d}x \right) }_{p_k} &= \frac{1}{2} \, \sum_{k \in \mathbb{Z}} p_k \, q_k \end{align}}

(since we are deriving the Daubechies wavelet, we will 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 k \in \left[ 0, 3 \right]} , but this could be arbitrary.)

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 \sum_{k \in \mathbb{Z}} p_k \, q_k = \sum_{k = 0}^3 p_k \, q_k}

We want 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 q_0, q_1, q_2, q_3 = q_{-2}, q_{-1}, q_0, q_1} ?

We get $q_{k}=\left(-1\right)^{k}\,p_{1-k}$, and we check 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 \sum_{k} q_k \, q_{k-2\ell} = 2 \delta_{0, \ell}}
• $\sum _{k}p_{k}=2$
• 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 \sum_{k} q_k = 0}

Note: In the Haar wavelet, we have 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 p_0 {{=}} 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 p_1 {{=}} 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 q_0 {{=}} 1} , and 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 q_1 {{=}} -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 q_{-2} = \frac{1-\sqrt{3}}{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 q_{-1} = \frac{\sqrt{3}-3}{4}} , 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 q_0 = \frac{3 + \sqrt{3}}{4}} , 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 q_1 = \frac{\sqrt{3}-1}{4}}

So 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 W_0} , we get 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 \psi(x) = \frac{1-\sqrt{3}}{4} \, \phi(2x+2) + \frac{\sqrt{3}-3}{4} \, \phi(2x+1) + \frac{3 + \sqrt{3}}{4} \, \phi(2x) - \frac{1-\sqrt{3}}{4} \, \phi(2x-1)} , and 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) \in V_1} .