Mathematical Framework of Compactly Supported Orthogonal Wavelets

Abstract

After the wavelet method for processing the jumping signal was proposed in the 1970s, its powerful advantages attract the attention of mathematicians, and the mathematical framework of a wavelet has been gradually established and various wavelets with different anticipatable properties are constructed since 1980s.

Appendices

Appendix 2.1 Moment Relationships of Orthogonal Scaling Functions

From the description of Property 2.2, we know that the expression

$$M_{i} = \frac{1}{2}\sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} k^{i} \, (i = 0,1,2, \ldots ,r)}$$

and Property 2.2 should be proved, which will give in Appendix 2.2.

According to Definition 2.4 of the moment of the scaling function, we have

$$M_{i} = \int\limits_{ - \infty }^{\infty } {x^{i} \varphi (x)} dx = \int\limits_{0}^{{\tilde{N}}} {x^{i} \varphi (x)} dx\quad \quad 0 \le i \le s.$$

(2.107)

where $M_{0} = 1$ is prechosen in the establishing wavelet. Substituting Eq. (2.2) into Eq. (2.107), one gets

$$\begin{aligned} M_{i} & = \sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} } \int\limits_{ - \infty }^{\infty } {x^{i} \varphi (2x - k)dx} = \frac{1}{2}\sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} } \int\limits_{ - \infty }^{\infty } {\left( {\frac{x + k}{2}} \right)^{i} \varphi (x)dx} \\ & = \frac{1}{{2^{i + 1} }}\sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} } \sum\limits_{j = 0}^{i} {C_{i}^{j} k^{j} } \int\limits_{ - \infty }^{\infty } {x^{i - j} \varphi (x)dx} = \frac{1}{{2^{i + 1} }}\sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} } \sum\limits_{j = 0}^{i} {C_{i}^{j} k^{j} } M_{i - j} \\ \end{aligned}$$

(2.108)

in which $C_{i}^{j} = \frac{j!(i - j)!}{i!}$ is the expansion coefficients of $(x + k)^{i}$. Denote

$$m_{j} = \frac{1}{2}\sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} k^{j} } .$$

(2.109)

and consider $C_{i}^{0} = 1 = k^{0}$, then, Eq. (2.108) can be rewritten by the form

$$M_{i} = \frac{1}{{2^{i} - 1}}\sum\limits_{j = 1}^{i} {C_{i}^{j} m_{j} } M_{i - j} .$$

(2.110)

Appendix 2.2 Moment Relationships of Scaling Functions of Coiflets

Conclusion 2.1. To a compactly supported orthogonal wavelet with up to s-order vanishing moment, when the relationships

$$M_{j} = M_{1}^{j} ,\quad j = 1,2, \ldots ,s.$$

(2.111)

are established, then we have

$$\left( {x - M_{1} } \right)^{j} = \sum\limits_{{k \in {\text{Z}}}} {k^{j} \varphi (x - k)} ,\quad j = 1,2, \ldots ,s,$$

(2.112)

$$M_{i} = m_{i} ,\quad i = 1,2, \ldots ,s.$$

(2.113)

Proof

• (I) To prove Eq. (2.112). Since the wavelet has up to s-order vanishing moment, following Property 2.1, we have

$$\left( {x - M_{1} } \right)^{j} \equiv \sum\limits_{k \in Z} {d_{k}^{j} \varphi (x - k)} \quad ,\quad x \in R,\quad 0\le \, j \le s,$$

(2.114)

$$d_{k}^{j} = \int\limits_{ - \infty }^{\infty } {(x - M_{1} )^{j} \varphi (x - k)dx} .$$

(2.115)

Taking the transform of the integral variable to Eq. (2.115), one gets

$$\begin{aligned} d_{k}^{j} & = \int\limits_{ - \infty }^{\infty } {(x + k - M_{1} )^{j} \varphi (x)dx} \\ & = \sum\limits_{i = 0}^{j} {C_{j}^{i} k^{i} } \int\limits_{ - \infty }^{\infty } {(x - M_{1} )^{j - i} \varphi (x)dx} . \\ \end{aligned}$$

(2.116)

Since $C_{j}^{j} = 1$ and $\int_{ - \infty }^{\infty } \varphi (x)dx = 1$, Eq. (2.116) can be reduced into the form:

$$\begin{aligned} d_{k}^{j} & = k^{j} + \sum\limits_{i = 0}^{j - 1} {C_{j}^{i} k^{i} } \int\limits_{ - \infty }^{\infty } {(x - M_{1} )^{j - i} \varphi (x)dx} \\ & = k^{j} + \sum\limits_{i = 0}^{j - 1} {C_{j}^{i} k^{i} } \sum\limits_{l = 0}^{j - i} {C_{j - i}^{l} ( - 1)^{l} } M_{1}^{l} \int\limits_{ - \infty }^{\infty } {x^{j - i - l} \varphi (x)dx} . \\ & = k^{j} + \sum\limits_{i = 0}^{j - 1} {C_{j}^{i} k^{i} } \sum\limits_{l = 0}^{j - i} {C_{j - i}^{l} ( - 1)^{l} } M_{1}^{l} M_{j - i - l} \\ \end{aligned}$$

(2.117)

Substitution Eq. (2.111) into Eq. (2.117) leads to

$$\begin{aligned} d_{k} & = k^{j} + \sum\limits_{i = 0}^{j - 1} {C_{j}^{i} k^{i} } \sum\limits_{l = 0}^{j - i} {C_{j - i}^{l} ( - 1)^{l} } M_{1}^{j - i} \\ & = k^{j} + \sum\limits_{i = 0}^{j - 1} {C_{j}^{i} k^{i} M_{1}^{j - i} } \sum\limits_{l = 0}^{j - i} {C_{j - i}^{l} ( - 1)^{l} } . \\ & = k^{j} + \sum\limits_{i = 0}^{j - 1} {C_{j}^{i} k^{i} M_{1}^{j - i} } (1 - 1)^{j - i} = k^{j} \\ \end{aligned}$$

(2.118)

Then, substituting the above result of Eq. (2.118) into Eq. (2.114), we get Eq. (2.112).

• (II) To prove Eq. (2.113). Here, we use mathematical induction for this proof. When $i = 1$, according to Eq. (2.110), it is obvious that $M_{1} = \frac{1}{{2^{1} - 1}}C_{1}^{1} m_{1} M_{0} = m_{1}$ is held since $C_{1}^{1} = 1 = M_{0}$. Hence, Eq. (2.113) holds for $i = 1$. Assume Eq. (2.113) holds for $i = r(1 \le r < s)$, that is, the equations

$$M_{i} = m_{i} \quad ,\quad i = 1,2, \ldots ,r$$

(2.119)

are held. For the case of $i = r + 1$, Eq. (2.110) can be expressed by

$$M_{r + 1} = \frac{1}{{2^{r + 1} - 1}}\sum\limits_{j = 1}^{r + 1} {C_{r + 1}^{j} m_{j} M_{r + 1 - j} } = \frac{1}{{2^{r + 1} - 1}}\left\{ {m_{r} + \sum\limits_{j = 1}^{r} {C_{r + 1}^{j} m_{j} M_{r + 1 - j} } } \right\}.$$

(2.120)

Substitution of Eq. (2.119) into Eq. (2.120) leads to

$$M_{r + 1} = \frac{1}{{2^{r + 1} - 1}}\left\{ {m_{r + 1} + \sum\limits_{j = 1}^{r} {C_{r + 1}^{j} M_{j} M_{r + 1 - j} } } \right\}.$$

(2.121)

According to the conditions of Eqs. (2.111) and (2.121) is reduced in the form

$$M_{r + 1} = \frac{1}{{2^{r + 1} - 1}}\left\{ {m_{r + 1} + \sum\limits_{j = 1}^{r} {C_{r + 1}^{j} M_{1}^{r + 1} } } \right\} = \frac{1}{{2^{r + 1} - 1}}\left\{ {m_{r + 1} + M_{r + 1} \left( {\sum\limits_{j = 0}^{r + 1} {C_{r + 1}^{j} - 2} } \right)} \right\}$$

Since $\sum\limits_{j = 0}^{r} {C_{r + 1}^{j} } = (1 + 1)^{r + 1} = 2^{r + 1}$, the above equation becomes into

$$M_{r + 1} = \frac{1}{{2^{r + 1} - 1}}\left\{ {m_{r + 1} + M_{r + 1} \left( {2^{r + 1} - 2} \right)} \right\}$$

Further, one gets $M_{r + 1} = m_{r + 1}$, i.e., Eq. (2.113) holds for $i = r + 1$. From the mathematical method of induction, we know that Eq. (2.113) holds for all integers of $1 \le i \le s$. Hence, the proof of conclusion (2.107) is finished.

Combining Eqs. (2.109) and (2.113), we have the expression of moments in terms of the filter coefficients $p_{k}$ of the form

$$m_{j} = \frac{1}{2}\sum\limits_{k \in Z} {p_{k} k^{j} = \frac{1}{2}\sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} k^{j} } } \quad ,\quad \quad j = 1,2, \ldots ,s$$

(2.122)

which is the first part of Property 2.2.

• (III) Proof of the second part in Property 2.2:

That is, when

$$M_{2j - 1} = \frac{1}{2}\sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} k^{2j - 1} } = M_{1}^{2j - 1} ,\quad \, j = 1,2, \ldots ,l( < [(r - 1)/2])$$

(2.123)

are prechosen in the establishment of a wavelet, then, we should prove the following equations:

$$M_{2j} = \frac{1}{2}\sum\limits_{k = 0}^{{\tilde{N}}} {p_{k} k^{2j} } = M_{1}^{2j} \quad ,\quad j = 1,2, \ldots ,l(2l \le r)$$

(2.124)

holding. Here, the mathematical method of induction is employed in this proof.

When $\, j = 1$, Eq. (2.124) becomes $M_{2} = \frac{1}{2}\sum\nolimits_{k = 0}^{{\tilde{N}}} {p_{k} k^{2} }$. According to Eqs. (2.114) and (2.115) and considering $c_{k} = \int_{ - \infty }^{\infty } {\varphi (x - k)dx} = \int_{ - \infty }^{\infty } {\varphi (x)dx} = 1$, we have

$$\sum\limits_{k \in Z} {\varphi (x - k)} = 1$$

(2.125)

$$\sum\limits_{k \in Z} {k\varphi (x - k)} = x - M_{1} .$$

(2.126)

Denote

$$\kappa_{k} = \int\limits_{ - \infty }^{\infty } {x\varphi (x)\varphi (x - k)dx} .$$

(2.127)

Then

$$\begin{aligned} \kappa_{ - k} & = \int\limits_{ - \infty }^{\infty } {x\varphi (x)\varphi (x + k)dx} = \int\limits_{ - \infty }^{\infty } {(x - k)\varphi (x - k)\varphi (x)dx} \\ & = \int\limits_{ - \infty }^{\infty } {x\varphi (x - k)\varphi (x)dx} - k\int\limits_{ - \infty }^{\infty } {\varphi (x - k)\varphi (x)dx} . \\ & = \int\limits_{ - \infty }^{\infty } {x\varphi (x - k)\varphi (x)dx} - k\delta_{0,k} = \kappa_{k} \\ \end{aligned}$$

(2.128)

Thus, we have $\sum\nolimits_{k \in Z} {k\kappa_{k} } = \sum\nolimits_{k = 1}^{\infty } {k\kappa_{k} } + \sum\nolimits_{k = - 1}^{ - \infty } {k\kappa_{k} } = 0$. Multiplying $x\varphi (x)$ to Eq. (2.126), and integrating the induced equation respective to x in $( - \infty ,\infty )$, one obtains

$$\begin{aligned} 0 &= \sum\limits_{k \in Z} {k\kappa_{k} } = \sum\limits_{k \in Z} k \int\limits_{ - \infty }^{\infty } {x\varphi (x)\varphi (x - k)dx}\\ & = \int\limits_{ - \infty }^{\infty } {x^{2} \varphi (x)dx} - M_{1} \int\limits_{ - \infty }^{\infty } {x\varphi (x)dx} = M_{2} - M_{1}^{2} . \end{aligned}$$

Thus, Eq. (2.124) holds for $\, j = 1$.

When $j = p \, ( < l)$, we assume that the following equations:

$$M_{2j} = M_{1}^{2j} ,\quad \quad j = 2,3, \ldots ,p$$

(2.129)

are held. Then for $j = p + 1$, the equation $M_{2(p + 1)} = M_{1}^{2(p + 1)}$ should be proved. According to Eq. (2.112), we have

$$(x - M_{1} )^{2p + 1} = \sum\limits_{k \in Z} {k^{2p + 1} \varphi (x - k)} .$$

(2.130)

Taking the calculations of Eq. (2.130) as same as ones of Eq. (2.126), we get

$$\int\limits_{ - \infty }^{\infty } {\left( {x - M_{1} } \right)^{2p + 1} x\varphi (x)dx} = \sum\limits_{k \in Z} k^{2p + 1} \int\limits_{ - \infty }^{\infty } {x\varphi (x)\varphi (x - k)dx} = \sum\limits_{k \in Z} k^{2p + 1} \kappa_{k} = 0.$$

(2.131)

To the left side of Eq. (2.131), we have

$$\begin{aligned} \int\limits_{ - \infty }^{\infty } {\left( {x - M_{1} } \right)^{2p + 1} x\varphi (x)dx} & = \sum\limits_{j = 0}^{2p + 1} {C_{2p + 1}^{j} } ( - 1)^{j} M_{1}^{j} \int\limits_{ - \infty }^{\infty } {x^{2p - j + 2} \varphi (x)dx} \\ & = \sum\limits_{j = 0}^{2p + 1} {C_{2p + 1}^{j} } ( - 1)^{j} M_{1}^{j} M_{2p - j + 2}\\ & = M_{2p + 2} + \sum\limits_{j = 1}^{2p + 1} {C_{2p + 1}^{j} } ( - 1)^{j} M_{1}^{j} M_{2p - j + 2} . \\ \end{aligned}$$

(2.132)

Since $2p + 2 - j \le 2p + 1$ $(1 < j \le 2p + 1)$ in the summation of Eq. (2.132), from Eqs. (2.123) and (2.129), one gets

$$M_{2p - j + 1} = M_{1}^{2p - j + 1} ,\quad \quad j = 1,2, \ldots ,2p + 1.$$

(2.133)

Substitution of Eq. (2.133) into (2.132) yields

$$\begin{aligned} \int\limits_{ - \infty }^{\infty } {\left( {x - M_{1} } \right)^{2p + 1} x\varphi (x)dx} & = M_{2p + 2} + M_{1}^{2p + 2} \sum\limits_{j = 1}^{2p + 1} {C_{2p + 1}^{j} } ( - 1)^{j} \\ & = M_{2p + 2} - M_{1}^{2p + 2} + M_{1}^{2p + 2} \sum\limits_{j = 0}^{2p + 1} {C_{2p + 1}^{j} } ( - 1)^{j} \\ & = M_{2p + 2} - M_{1}^{2p + 2} . \\ \end{aligned}$$

(2.134)

Substituting Eq. (2.146) into Eq. (2.131), one gets

$$M_{2p + 2} = M_{1}^{2p + 2} .$$

According to the mathematical method of induction, we know that Eq. (2.124) holds for all integers of $j = 1,2, \ldots ,l(2l \le r)$. Thus, the proof of Property 2.2 is finished.

Appendix 2.3 Condition on Filer Coefficients from Vanishing Moments

In Definition 2.5, we give the general expression of vanishing moments of a compact supported orthogonal wavelet, see Eq. (2.25). Here, we give their explicit expressions in terms of the filter coefficients $p_{k} (k = 0,1,2, \ldots ,\tilde{N})$. Substituting Eq. (2.15b) into Eq. (2.25), one gets

$$\begin{aligned} 0 & = \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } \int\limits_{R} {x^{i} \varphi (2x - k)} dx = \frac{1}{{2^{i + 1} }}\sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } \int\limits_{R} {(x + k)^{i} \varphi (x)} dx \\ & = \frac{1}{{2^{i + 1} }}\sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } \sum\limits_{l = 0}^{i} {C_{i}^{l} } k^{l} \int\limits_{R} {x^{i - l} \varphi (x)} dx = \frac{1}{{2^{i + 1} }}\sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } \sum\limits_{l = 0}^{i} {C_{i}^{l} } k^{l} M^{i - l} \\ & = \frac{1}{{2^{i + 1} }}\sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } \sum\limits_{l = 0}^{i} {C_{i}^{l} } k^{l} M_{1}^{i - l} \\ \end{aligned}$$

i.e.,

$$\sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } \sum\limits_{l = 0}^{i} {C_{i}^{l} } k^{i} M_{1}^{i - l} = 0\quad ,\quad i = 0,1, \ldots ,r - 1.$$

(2.135)

When $i = 0$, we have

$$\sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } = 0.$$

(2.136)

Further, when $i = 1$ and Eq. (2.136) is considered, Eq. (2.135) becomes

$$\begin{aligned} & \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } (C_{1}^{0} k^{0} M_{1}^{1} + C_{1}^{1} k^{1} M_{1}^{0} ) \\ & = \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } + \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k = \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k = 0 \\ \end{aligned}$$

i.e., $\sum\nolimits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k = 0$. Assume $\sum\nolimits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k^{i} = 0$ held for $i = 2, \ldots ,s( < r - 1)$. Then, for $i = s + 1( \le r)$, Eq. (2.135) can be rewritten in the form:

$$\begin{aligned} 0 & = \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } \sum\limits_{l = 0}^{s + 1} {C_{s + 1}^{l} } k^{l} M_{1}^{s + 1 - l} \\ & = \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } (M_{1}^{s + 1} + C_{s + 1}^{1} kM_{1}^{s} + C_{s + 1}^{2} k^{2} M_{1}^{s - 1} + \cdots + C_{s + 1}^{s} k^{s} M_{1}^{1} + k^{s + 1} M_{1}^{0} ) \\ & = M_{1}^{s + 1} \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } + C_{s + 1}^{1} M_{1}^{s} \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k + C_{s + 1}^{2} M_{1}^{s - 1} \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k^{2} + \cdots \\ & \quad + C_{s + 1}^{s} M_{1}^{{}} \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k^{s} + \sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k^{s + 1} . \\ \end{aligned}$$

(2.137)

According to the previous assumption and results, we know that

$$\sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k^{i} = 0\quad ,\quad i = 0,1, \ldots ,s$$

(2.138)

are held. Substitution such results of Eq. (2.138) into Eq. (2.137) yields

$$\sum\limits_{k = 0}^{{\tilde{N}}} {( - 1)^{k} p_{k} } k^{s + 1} = 0$$

being held too. By means of the mathematical induction, we get that Eq. (2.138) is true for arbitrary integer $i = 0,1, \ldots ,r - 1$.