MATH 414 Lecture 38

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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) = \sum_{k} p_k \, \phi(2x-k)}

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 \hat{\phi}(\xi) = P \left( \mathrm{e}^{-\frac{i\,\xi}{2}} \right) \, \hat{\phi} \left( \frac{\xi}{2} \right)} , 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(z) = \frac{1}{2} \, \sum_k p_k \, z^k}

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 \hat{\phi}(\xi) = \frac{1}{\sqrt{2\pi}} \, \prod_{r=1}^\infty P \left( \mathrm{e}^{-\frac{i\,\xi}{2^r}} \right)}

• This means we can start 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 p_k} 's and 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 \hat{\phi}(\xi)} and $\phi (x)={\mathcal {F}}^{-1}\left[{\hat {\phi }}\right]$
• $p_{k}$'s satisfy at least the following properties:
  • $\sum _{k}p_{k-2\ell }\,p_{k}=2\delta _{\ell ,0}$
  • $\sum _{k}p_{k}=2$
  • $\sum _{k}p_{2k}=1$ and $\sum _{k}p_{2k+1}=1$.

Wavelet

${\hat {\psi }}(\xi )=-\mathrm {e} ^{-{\frac {i\,\xi }{2}}}\,P\left(-\mathrm {e} ^{\frac {i\,\xi }{2}}\right)\,{\hat {\phi }}\left({\frac {\xi }{2}}\right)$

${\hat {\psi }}$ is known!

Haar Example

$P(z)={\frac {1}{2}}\,\left(1+z\right)$

${\begin{aligned}v_{N}&=\prod _{r=1}^{N}{\frac {1}{2}}\,\left(1+\mathrm {e} ^{-{\frac {i\,\xi }{2^{r}}}}\right)\\&=\left({\frac {1+\mathrm {e} ^{-{\frac {i\,\xi }{2^{N}}}}}{2}}\right)\,\left(1+\mathrm {e} ^{-{\frac {i\,\xi }{2^{N-1}}}}\right)\dots \left(1+\mathrm {e} ^{-{\frac {i\,\xi }{2}}}\right)\end{aligned}}$

Let $z=\mathrm {e} ^{-{\frac {i\,\xi }{2^{N}}}}$. Then

${\begin{aligned}v_{N}&=\left({\frac {1+z}{2}}\right)\,\left({\frac {1+z}{2}}\right)\,\left({\frac {1+z^{4}}{2}}\right)\dots \left({\frac {1+z^{2^{N-1}}}{2}}\right)\\(1-z)\,v_{N}&={\frac {1}{2^{N}}}\,(1-z^{2n})\\&={\frac {1}{2^{N}}}\,\left(1-\mathrm {e} ^{-i\,\xi }\right)\\v_{N}&={\frac {1}{1-\mathrm {e} ^{-{\frac {i\,\xi }{2^{N}}}}}}\,\left(1-\mathrm {e} ^{-i\,\xi }\right)\,{\frac {1}{2^{N}}}\end{aligned}}$

We take a power series expansion of $\mathrm {e} ^{x}=1+x+{\frac {x^{2}}{2}}+\dots$

${\begin{aligned}1-\mathrm {e} ^{-{\frac {i\,\xi }{2^{N}}}}&=\left({\frac {i\,\xi }{2^{N}}}\right)+{\frac {(\dots )}{2^{2N}}}+\dots \\2^{N}\,\left(1-\mathrm {e} ^{-{\frac {i\,\xi }{2^{N}}}}\right)&=i\,\xi +O(2^{-N})\end{aligned}}$

The limit as $N\to \infty$ gives

$V_{N}=\dots ={\frac {2\sin {\left({\frac {\xi }{2}}\right)}}{\xi }}$

Hence ${\hat {\phi }}(\xi )={\frac {1}{\sqrt {2\pi }}}\,{\frac {2}{\xi }}\,\sin {\left({\frac {\xi }{2}}\right)}={\mathcal {F}}\left[\phi _{haar}\right]$

${\hat {\phi }}(\xi )={\frac {1}{\sqrt {2\pi }}}\,\prod _{r=1}^{\infty }P\left(\mathrm {e} ^{-{\frac {i\,\xi }{2^{r}}}}\right)$, where ${\hat {\phi }}_{N}$ is a partial product.

When ${\hat {\phi }}$ satisfies an iterative relation:

• $\phi _{0}(x)=\phi _{haar}(x)$
• $\phi _{N}(x)=\sum _{k}p_{k}\,\phi _{N-1}(2x-k)$

Theorem. Suppose $P(z)={\frac {1}{2}}\,\sum _{k}p_{k}\,z^{k}$, where the $p_{k}$'s are given. If:

1. $\left|P(z)\right|^{2}+\left|P(-z)\right|^{2}=1$, $\left|z\right|=1$, where $z=\mathrm {e} ^{i\,t}$
2. $P(1)=1$
3. $\left|P\left(\mathrm {e} ^{i\,t}\right)\right|>0$ for $\left|t\right|\leq {\frac {\pi }{2}}$

Then ${\frac {1}{\sqrt {2\pi }}}\,\prod _{r=1}^{N}P\left(\mathrm {e} ^{-{\frac {i\,\xi }{2}}}\right)$ converges in $L^{2}$ to a function ${\hat {\phi }}(\xi )$.

Proof.

quod erat demonstrandum

Point:

1. $\phi (x)={\mathcal {F}}^{-1}\left[{\hat {\phi }}(\xi )\right]$ is a scaling function
2. ...

Moments of Wavelet

A moment of a function $\psi (x)$ is an integral of the form:

$M_{k}=\int _{-\infty }^{\infty }x^{k}\,\psi (x)\,\mathrm {d} x$, where $k=0,1,\dots$

Recall that if $M_{0}=0$ and $M_{1}=0$, then for the Daubechies ( $N=2$ ) wavelet (projection of $f(x)$ onto $\psi _{jk}$ component of 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} space):

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} b_k^{j-1} &= \left\langle f, \psi_{jk} \right\rangle = \left\langle f(x), \psi \left( 2^j \, x-k \right) \right\rangle \\ &= \int_{-\infty}^\infty f(x) \, \psi \left( 2^j \, x - k \right) \, \mathrm{d}x \\ &= \int_{-\infty}^{\infty} f \left( 2^{-j} \, k + 2^{-j} \, u \right) \, \psi(u) \, \mathrm{d}u \end{align}}

We can take the Taylor series expansion of 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} centered at 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} \, u}

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 \left( 2^{-j} \, k + 2^{-j} \, u \right) = f \left( 2^{-j} \, k \right) + f' \left( 2^{-j} \, k \right) \, 2^{-j} \, u + \frac{f'' \left( 2^{-j} \, k \right)}{2} \, \left( 2^{-j} \, u \right)^2 + \dots}

... and plug it into the inner product projection 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 b_k^{j-1}} above:

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} b_k^{j-1} &= f \left( 2^{-j} \, k \right) \, \int_{-\infty}^{\infty} \psi(x) \, \mathrm{d}x + f' \left( 2^{-j} \, k \right) \, 2^{-j} \, u \, \int_{-\infty}^{\infty} u \, \psi(x) \, \mathrm{d}x + f'' \left( 2^{-j} \, k \right) \, 2^{-2j} \, \int_{-\infty}^{\infty} u^2 \, \psi(x) \, \mathrm{d}x + \dots \\ &\approx \frac{f'' \left( 2^{-j} \, k \right) \, 2^{-2j}}{2} \, M_2 \end{align}}

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 much greater than 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 f} is smooth, 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 b_k^{j-1}} is small.

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'} is discontinuous 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 k \le u < k+1} , then coefficients can become large.

Hence we can detect singularities (discontinuities) 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 f'} because 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 b_k^{j-1}} will be relatively large near 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 x_{sing} = 2^{-j} \, k}

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 M_k = 0} 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 k = 0, \ldots, N-1} , then for smooth 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} ,

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 b_k^{j-1} \approx \frac{f^{(N)} \left( 2^{-j} \, k \right)}{N!} \, 2^{-N \, j} \, M_N + \dots}

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(\xi) = -\mathrm{e}^{ -\frac{i\,\xi}{2} } \, P \left( -\mathrm{e}^{ \frac{i \, \xi}{2} } \right) \, \hat{\phi} \left( \frac{\xi}{2} \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 M_k = \int_{-\infty}^{\infty} x^k \, \psi(x) \,\mathrm{d}x = \sqrt{2\pi} i^k \frac{\mathrm{d}^k \, \hat{\psi}}{\mathrm{d}\xi^k}}

Vanishing Moments

Demand: 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(z) = \left( \frac{1+z}{2} \right)^N \, \tilde{P}(z)} , 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 \tilde{P}(-1) \ne 0}

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 \hat{\psi}(\xi) = \sin^N \left( \frac{\xi}{4} \right) \, F \left( \mathrm{e}^{ \frac{i \, \xi}{2}} \right) \sim \xi^N \, F(\dots)}

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. \frac{\mathrm{d}^k\hat{\psi}}{\mathrm{d}\xi^k} \right|_{\xi=0} = 0} implies 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_k = 0}

Require: 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(z) = \frac{1}{2} \, \left( p_0 + p_1 \, z + p_2 \, z^2 + p_3 \, z^3 \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 P(z) = \frac{ \left( 1+z \right)^2}{2} \, \left( a \, z + b \right)} , 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 \tilde{P} (z) = a \, z + b}

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} \hat{\psi}(\xi) &= -\mathrm{e}^{-\frac{i\,\xi}{2}} \, \left( \frac{1-\mathrm{e}^{ \frac{i\,\xi}{2} }}{2} \right)^2 \, \hat{\phi} \left( \frac{\xi}{2} \right) \\ &= (-i)^2 \, \sin^2 \left( \frac{\xi}{4} \right) \, \hat{\phi} \left( \frac{\xi}{2} \right) \\ &= \sin^2 \left( \frac{\xi}{4} \right) \, \tilde{P} \left( -\mathrm{e}^{ \frac{i \, \i}{2}} \right) \, \hat{\phi} \left( \frac{\xi}{2} \right) \\ &= \dots \end{align}}

Going back to conditions 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 P(z)} ...

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) = \left( \frac{2}{2} \right)^2 \, (a + b) = a + b = 1} , 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(z) = \left( \frac{1+z}{2} \right)^2 \left( b \, z + 1 - b \right)}

Satisfy: 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| P(z) \right|^2 + \left| P(-z) \right|^2 = 1} , 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 z = \mathrm{e}^{i \, \theta}} .

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 \cos^4 \left( \frac{\theta}{2} \right) \, \left| b \, \mathrm{e}^{i \, \theta} + 1-b \right|^2 + \sin^4 \left( \frac{\theta}{2} \right) \, \left| -b \, \mathrm{e}^{i\,\theta} + 1-b \right|^2 = 1}

Use trig identities until you're sick in the face... What you get when all is said and done is:

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^2 + \left( 1-a \right)^2 &= 4 \\ 2a^2 - 2a + 1 &= 4 \\ a^2 - a - 3 &= 0 \\ a &= \frac{1 \pm \sqrt{3}}{4} \end{align}}

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} 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 b} form 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 for the Daubechies wavelet