MATH 414 Lecture 38

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Scaling Function

$\phi (x)=\sum _{k}p_{k}\,\phi (2x-k)$

${\hat {\phi }}(\xi )=P\left(\mathrm {e} ^{-{\frac {i\,\xi }{2}}}\right)\,{\hat {\phi }}\left({\frac {\xi }{2}}\right)$, where $P(z)={\frac {1}{2}}\,\sum _{k}p_{k}\,z^{k}$

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

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

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 \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)}

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}} is known!

Haar Example

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( 1 + z \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 \begin{align} 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{align}}

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 z = \mathrm{e}^{-\frac{i \, \xi}{2^N}}} . 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 \begin{align} 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{align}}

We take a power 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 \mathrm{e}^{x} = 1 + x + \frac{x^2}{2} + \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 \begin{align} 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{align}}

The limit 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 N \to \infty} gives

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_N = \dots = \frac{2\sin{ \left( \frac{\xi}{2} \right) }}{\xi}}

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}} \, \frac{2}{\xi} \, \sin{ \left( \frac{\xi}{2} \right) } = \mathcal{F}\left[ \phi_{haar} \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 \hat{\phi}(\xi) = \frac{1}{\sqrt{2\pi}} \, \prod_{r=1}^{\infty} P \left( \mathrm{e}^{-\frac{i\,\xi}{2^r}} \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 \hat{\phi}_N} is a partial product.

When 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}} satisfies an iterative relation:

• 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_0(x) = \phi_{haar}(x)}
• 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_N(x) = \sum_k p_k \, \phi_{N-1}(2x-k)}

Theorem. Suppose 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} , where 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 are given. If:

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 \left| P(z) \right|^2 + \left| P(-z) \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 \left| z \right| = 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\,t}}
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 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 \left| P \left( \mathrm{e}^{i\,t} \right) \right| > 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 \left| t \right| \le \frac{\pi}{2}}

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

Proof.

quod erat demonstrandum

Point:

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

Moments of Wavelet

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

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} , 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 k = 0, 1, \dots}

Recall that 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_0 = 0} 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 M_1 = 0} , then for the 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 N = 2} ) wavelet (projection 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(x)} onto 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_{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