MATH 414 Lecture 39

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MRA in terms of Fourier Transforms

Gives (yet another) way to construct 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 V_j} spaces, scaling function, and wavelet function: by finding Fourier transforms that depend on a special polynomial $P(z)$.

Scaling Function

Recall the scaling relation:

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

Theorem 5.19. The Fourier Transform of a MRA 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)} is given by:

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) = \hat{\phi} \left( \frac{\xi}{2} \right) \, P \left( \mathrm{e}^{-\frac{i\,\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) = \sum_{k \in \mathbb{Z}} p_k \, z^k}

Proof. Use the scaling relation and follow the definitions.

Evaluating the Fourier Transform 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 \phi(x) = \sum_{k \in \mathbb{Z}} p_k \, \phi(2x-k)} (with a change of 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} ) 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 \begin{align} \mathcal{F} \left[ \phi(x) \right] (\xi) = \hat{\phi}(\xi) &= \sum_{k \in \mathbb{Z}} p_k \, \mathcal{F} \left[ \phi(2x-k) \right](\xi) && \mbox{where}\ \mathcal{F} \left[ \phi(2x-k) \right] (\xi) = \frac{1}{2} \, \mathrm{e}^{-\frac{i \, \xi \, k}{2}} \, \hat{\phi} \left( \frac{\xi}{2} \right) \\ &= \sum_{k \in \mathbb{Z}} p_k \, \left( \frac{1}{2} \, \mathrm{e}^{-\frac{i \, \xi \, k}{2}} \, \hat{\phi} \left( \frac{\xi}{2} \right) \right) \\ &= \hat{\phi} \left( \frac{\xi}{2} \right) \, \frac{1}{2} \, \sum_{k \in \mathbb{Z}} p_k \, \mathrm{e}^{-\frac{i \, \xi \, k}{2}} && \mbox{let}\ P(z) = \frac{1}{2} \, \sum_{k \in \mathbb{Z}} p_k \, z^k \\ &= \hat{\phi} \left( \frac{\xi}{2} \right) \, P \left( \mathrm{e}^{-\frac{i \, \xi}{2}} \right) \end{align}}

quod erat demonstrandum

Corollary. 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 \phi(x)} satisfies the normalization condition: 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) \, \mathrm{d}x = 1} , 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 \hat{\phi}(\xi) = \frac{1}{\sqrt{2\pi}} \, \prod_{r=1}^\infty P \left( \mathrm{e}^{-\frac{i\,\xi}{2^r}} \right)}

Proof. Evaluating the recursive function (from the theorem 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 N} times 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 \begin{align} \hat{\phi}(\xi) &= \hat{\phi} \left( \frac{\xi}{2^N} \right) \, P \left( \mathrm{e}^{-\frac{i\,\xi}{2}} \right) \, P \left( \mathrm{e}^{-\frac{i\,\xi}{4}} \right) \, P \left( \mathrm{e}^{-\frac{i\,\xi}{8}} \right) \, \dots \, P \left( \mathrm{e}^{-\frac{i\,\xi}{2^N}} \right) \\ &= \hat{\phi} \left( \frac{\xi}{2^N} \right) \, \prod_{r=1}^N P \left( \mathrm{e}^{-\frac{i\,\xi}{2^r}} \right) \end{align}}

Taking 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} tends to infinity 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 \begin{align} \hat{\phi}(\xi) &= \lim_{N \to \infty} \hat{\phi} \left( \frac{\xi}{2^N} \right) \, \prod_{r=1}^N P \left( \mathrm{e}^{-\frac{i \, \xi}{2^r}} \right) \\ &= \hat{\phi}(0) \, \prod_{r=1}^{\infty} P \left( \mathrm{e}^{-\frac{i\,\xi}{2^r}} \right) \end{align}}

...and we know what 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}(0)} 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} \hat{\phi}(0) &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} \phi(x) \, \mathrm{e}^{-i \, (0) \, x} \, \mathrm{d}x \\ &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^\infty \phi(x) \, \mathrm{d}x \\ &= \frac{1}{\sqrt{2\pi}} \end{align}}

Therefore

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)}

quod erat demonstrandum

Wavelet Function

We can do a similar thing for the wavelet 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)} given its definition in terms 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 \phi(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 \psi(x) = \sum_{k \in \mathbb{Z}} (-1)^k \, \overline{p_{1-k}} \, \phi(2x-k)}

Theorem. The Fourier Transform of a MRA wavelet 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 given by:

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

---

Moreover, assuming 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 \in \mathbb{R}} 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 \in \mathbb{Z}} , the above is equivalent to

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) = -z \, \hat{\phi} \left( \frac{\xi}{2} \right) \, P \left( -\bar{z} \right)}

Proof. Since 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}} (-1)^k \, \overline{p_{1-k}} \, \phi(2x-k)} , the Fourier transform 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 \psi(x)} 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} \mathcal{F} \left[ \psi(x) \right] (\xi) &= \sum_{k \in \mathbb{Z}} (-1)^k \, \overline{p_{1-k}} \, \mathcal{F} \left[ \phi(2x-k) \right](\xi) \\ &= \sum_{k \in \mathbb{Z}} (-1)^k \, \overline{p_{1-k}} \, \left( \frac{1}{2} \, \mathrm{e}^{-\frac{i \, \xi \, k}{2}} \, \hat{\phi} \left( \frac{\xi}{2} \right) \right) \\ &= \hat{\phi} \left( \frac{\xi}{2} \right) \, \frac{1}{2} \, \sum_{k \in \mathbb{Z}} (-1)^k \, \overline{p_{1-k}} \, \mathrm{e}^{-\frac{i \, \xi \, k}{2}} && \ell = 1-k \\ &= \hat{\phi} \left( \frac{\xi}{2} \right) \, \left( -\frac{1}{2} \, \sum_{\ell \in \mathbb{Z}} (-1)^\ell \, \overline{p_{\ell}} \, \mathrm{e}^{-\frac{i \, \xi \, (1-\ell)}{2}} \right) \\ &= \hat{\phi} \left( \frac{\xi}{2} \right) \, \left( - \mathrm{e}^{-\frac{i \, \xi}{2}} \, \frac{1}{2} \, \sum_{\ell \in \mathbb{Z}} (-1)^\ell \, \overline{p_{\ell}} \, \mathrm{e}^{\frac{i\,\xi\,\ell}{2}} \right) && \mbox{let} \ z = \mathrm{e}^{-\frac{i\,\xi}{2}} \\ &= \hat{\phi} \left( \frac{\xi}{2} \right) \, \left( -z \, \frac{1}{2} \, \sum_{\ell \in \mathbb{Z}} (-1)^\ell \, \overline{p_{\ell}} \, \overline{z}^\ell \right) \\ &= -z \, \hat{\phi} \left( \frac{\xi}{2} \right) \, \left( \frac{1}{2} \, \sum_{\ell \in \mathbb{Z}} \overline{p_{\ell}} \, (-\overline{z})^\ell \right) \\ &= -z \, \hat{\phi} \left( \frac{\xi}{2} \right) \, \overline{P(-z)} \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 p_k} is real, 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 \overline{p_{\ell}} = p_{\ell}} , 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 \begin{align} \hat{\psi}(\xi) &= -z \, \hat{\phi} \left( \frac{\xi}{2} \right) \, \left( \frac{1}{2} \, \sum_{\ell \in \mathbb{Z}} p_{\ell} \, (-\overline{z})^\ell \right) \\ &= -z \, \hat{\phi} \left( \frac{\xi}{2} \right) \, P(-\overline{z}) \end{align}}

quod erat demonstrandum

Restrictions 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(x)}

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(z)} polynomial used above must satisfy a couple of conditions. These conditions directly correspond to the conditions we placed on 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} coefficients.

Theorem 5.23. Given the definition 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 P(z) = \frac{1}{2} \, \sum_{k \in \mathbb{Z}} p_k \, z^k} above, 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 P(z)} satisfies the following conditions

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} (since sum of 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 p_k} 's is 2)
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 \left| P(z) \right|^2 + \left| P(-z) \right|^2 = 1} 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 \left| z \right| = 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 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 \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 \hat{\phi}(\xi) = \frac{1}{\sqrt{2\pi}} \, \prod_{r=1}^{\infty} P \left( \mathrm{e}^{-\frac{i \, \xi}{2^r}} \right)}

is the Fourier transform of a scaling function.

Proof.

quod erat demonstrandum

Consequently, 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) = 0} since

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}} (-1)^k \, p_k = \sum_{k \in \mathbb{Z}} p_{2z} - \sum_{k \in \mathbb{Z}} p_{2z+1} = 1 - 1 = 0}

Note: Conditions 1 & 2 correspond to the following conditions on 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:

• 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 - 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 \in \mathbb{Z}} p_k {{=}} 2}

Daubechies Wavelet

Defined based on number 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} of vanishing moments. This is what makes this wavelet/MRA unique.

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 = 0} 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 < 0} or 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 > 2N-1} . This shall be used to construct 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_N(x)} .

What's really important 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} P(z) &= \frac{1}{2} \sum_{k \in \mathbb{Z}} p_k \, z^k \\ &= \frac{1}{2} \, \sum_{k = 0}^{2N-1} p_k \, z^k \\ &= \frac{1}{2} \, \left( p_0 + p_1 \, z + p_2 \, z^2 + \dots + p_{2N-1} \, z^{2N-1} \right) \end{align}}

This is just a polynomial of degree 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 2N-1} .

In regards to vanishing moments, recall that 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 k} ^th moment 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} is given by

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_{\mathbb{R}} x^k \, \psi(x) \, \mathrm{d}x}

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 \psi_N(x)} is defined to 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 N} vanishing moments: 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 = M_1 = \dots = M_{N-1} = 0}

In terms of the polynomial above, we factor as follows

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} P(z) &= \frac{1}{2} \, \left( p_0 + p_1 \, z + p_2 \, z^2 + \dots + p_{2N-1} \, z^{2N-1} \right) \\ &= \left( \frac{z+1}{2} \right)^N \, \tilde{P}(z) \end{align}}

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}(-1) \ne 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 \tilde{P}} has degree 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-1} .

Matlab Example

Suppose we wanted to find a wavelet decomposition 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 f(t) = 4t^2 + t - 1}

This is a degree 2 polynomial, so in order to have no 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^j} 's, we need a Daubechies wavelet 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 N} vanishing moments 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 N - 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 N = 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 f(t)} 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 db4} level 1 decomposition approximation 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_1(x) = \sum_{k \in \mathbb{Z}} a_k^1 \, \phi(2k-1)} .