Extended Wavelet Methods to 2D Irregular Domain and Local Refinement

Abstract

In the previous chapters, one can find that the definition domains for all the problems are restricted to regular ones when the wavelet-based method is employed, such as a rectangle in two-dimension and a cube in three-dimension.

Appendices

Appendix 11.1 Proof of Essential Properties of Interpolating Wavelet

In order to prove the relation of Eq. (11.8), we first investigate the nth moment $M_{n}^{\vartheta } = \int {x^{n} \vartheta (x)dx}$ of $\vartheta (x)$. Using the definition of Eq. (11.5) and the binomial theorem, we have

$$\begin{array}{*{20}l} {M_{n}^{\vartheta } = \int {x^{n} \vartheta (x)dx} = \int {x^{n} \int {\psi (y)\psi (y - x)dy} dx} = \int {\psi (y)[\int {(y - x)^{n} \psi (x)dx} ]dy} } \hfill \\ { = \int {\psi (y)[\int {\sum\limits_{i = 0}^{n} {C_{n}^{i} y^{n - i} ( - x)^{i} } \psi (x)dx} ]dy} = \sum\limits_{i = 0}^{n} {C_{n}^{i} ( - 1)^{i} M_{n - i}^{\psi } M_{i}^{\psi } } } \hfill \\ \end{array} .$$

(11.72)

Then following Eq. (11.3), one can obtain

$$M_{n}^{\vartheta } = 0,\;{\text{for}}\;n = 0,1, \ldots ,\gamma - 1.$$

(11.73)

From Eqs. (11.72) and (11.73), one can see that the auto-correlation function $\vartheta (x)$ of wavelet function has γ, and only has γ vanishing moments, because $M_{\gamma }^{\vartheta } = C_{\gamma }^{\gamma /2} ( - 1)^{\gamma /2} (M_{\gamma /2}^{\psi } )^{2} \ne 0$ since $M_{\gamma /2}^{\psi } \ne 0$ [12].

On the other hand, there is the relation $\vartheta (x) = 2\theta (2x) - \theta (x)$ [8, 14]. So, we have

$$M_{n}^{\vartheta } = \int {x^{n} [2\theta (2x) - \theta (x)]dx} = 2^{ - n} \int {(2x)^{n} \theta (2x)d(2x)} - \int {x^{n} \theta (x)dx} = (2^{ - n} - 1)M_{n}^{\theta } .$$

(11.74)

Considering Eq. (11.73), we obtain

$$M_{n}^{\theta } = 0,\;{\text{for}}\;n = 1,2, \ldots ,\gamma - 1.$$

(11.75)

We note that $M_{\gamma }^{\theta } \ne 0$ since $M_{\gamma }^{\vartheta } \ne 0$.

When m = 0, by using the definition of Eq. (11.5) and the property $M_{0}^{\phi } = 1$, one can directly obtain

$$M_{0}^{\theta } = \int {\theta (x)dx} = \int {\int {\phi (y)\phi (y - x)dy} dx} = \int {\phi (y) \left[ \int {\phi (y - x)dx} \right]dy} = 1.$$

(11.76)

Thus, the relation of Eq. (11.8) is proved.

For the relation of Eq. (11.9), considering the property $p_{2k} = \delta_{0,k}$ shown in Eq. (11.7), one can directly obtain

$$\sum\limits_{{k \in {\mathbb{Z}}}} {p_{2k} (2k)^{n} } = \delta_{0,n} .$$

(11.77)

On the other hand, by using the two-scale relation of Eq. (11.6) and the binomial theorem, we have

$$\begin{aligned} M_{n}^{\theta } & = \int {x^{n} [\theta (2x) + \sum\limits_{{k \in {\mathbb{Z}}}} {p_{2k + 1} \theta (2x - 2k - 1)} ]dx} \\ & = 2^{ - n - 1} \int {(2x)^{n} \theta (2x)d(2x)} + \frac{1}{2}\sum\limits_{{k \in {\mathbb{Z}}}} {p_{2k + 1} \int {(\frac{x + (2k + 1)}{2})^{n} \theta (x)dx} } \\ & = 2^{ - n - 1} M_{n}^{\theta } + 2^{ - n - 1} \sum\limits_{{k \in {\mathbb{Z}}}} {p_{2k + 1} \int {\sum\limits_{i = 0}^{n} {C_{n}^{i} (2k + 1)^{i} } x^{n - i} \theta (x)dx} } \\ & = 2^{ - n - 1} M_{n}^{\theta } + 2^{ - n - 1} \sum\limits_{{k \in {\mathbb{Z}}}} {p_{2k + 1} \sum\limits_{i = 0}^{n} {C_{n}^{i} (2k + 1)^{i} } M_{n - i}^{\theta } } . \\ \end{aligned}$$

(11.78)

Considering the relation of Eq. (11.8), for $n = 0,1, \ldots ,\gamma - 1$, Eq. (11.78) can be reduced into

$$M_{n}^{\theta } = 2^{ - n - 1} \delta_{0,n} + 2^{ - n - 1} \sum\limits_{k \in Z} {p_{2k + 1} (2k + 1)^{n} } = \delta_{0,n} .$$

(11.79)

According to Eqs. (11.77) and (11.79), the relation of Eq. (11.9) has been gained.

For the relation of Eq. (11.10), applying the binomial theorem, we get

$$\begin{aligned} g_{n,m} (x) & = \sum\limits_{{k \in {\mathbb{Z}}}} {(x - k)^{n} \theta^{(m)} (x - k)} \\ & = 2^{m} \sum\limits_{{k \in {\mathbb{Z}}}} {(x - k)^{n} \sum\limits_{{l \in {\mathbb{Z}}}} {p_{l} \theta^{(m)} (2x - 2k - l)} } \\ & = 2^{m} \sum\limits_{{k \in {\mathbb{Z}}}} {\sum\limits_{{i \in {\mathbb{Z}}}} {p_{i - 2k} (x - k)^{n} \theta^{(m)} (2x - i)} } \\ & = 2^{m - n} \sum\limits_{{k \in {\mathbb{Z}}}} {\sum\limits_{{i \in {\mathbb{Z}}}} {p_{i - 2k} [(2x - i) + (i - 2k)]^{n} \theta^{(m)} (2x - i)} } \\ & = 2^{m - n} \sum\limits_{{i \in {\mathbb{Z}}}} {\sum\limits_{o = 0}^{n} {C_{n}^{o} (2x - i)^{o} [\sum\limits_{{k \in {\mathbb{Z}}}} {p_{i - 2k} (i - 2k)^{n - o} } ]\theta^{(m)} (2x - i)} } . \\ \end{aligned}$$

(11.80)

Here, we have used the relation $\theta^{(m)} (x) = 2^{m} \sum\nolimits_{{k \in {\mathbb{Z}}}} {p_{k} \theta^{(m)} (2x - k)}$ which can be directly obtained from Eq. (11.6). Considering the relation in Eq. (11.9), then, Eq. (11.80) can be reduced into

$$g_{n,m} (x) = 2^{m - n} \sum\limits_{{i \in {\mathbb{Z}}}} {(2x - i)^{n} \theta^{(m)} (2x - i)} = 2^{m - n} g_{n,m} (2x).$$

(11.81)

Defining $G_{n,m} (x) = \int_{0}^{x} {g_{n,m} (y)dy}$ and following Eq. (11.81), we have

$$G_{n,m} (x) = 2^{m - n - 1} G_{n,m} (2x).$$

(11.82)

On the other hand, further, there is the relation

$$\begin{aligned}G_{n,m} (1) & = \int_{0}^{1} {\sum\limits_{{k \in {\mathbb{Z}}}} {(x - k)^{n} \theta^{(m)} (x - k)dx} } = \sum\limits_{{k \in {\mathbb{Z}}}} {\int_{ - k}^{ - k + 1} {x^{n} \theta^{(m)} (x)dx} } \\ & = \int {x^{n} \theta^{(m)} (x)dx} . \end{aligned}$$

(11.83)

For $n < m$, conducting n times integrations by parts to Eq. (11.83), we have

$$\begin{aligned} G_{n,m} (1) & = \int {x^{n} \theta^{(m)} (x)dx} \left. { = x^{n} \theta^{(m - 1)} (x)} \right|_{ - \infty }^{ + \infty } - n\int {x^{n - 1} \theta^{(m - 1)} (x)dx} \\ & = ( - 1)^{n} n!\int {\theta^{(m - n)} (x)} dx = ( - 1)^{n} n!\left. {\theta^{(m - n - 1)} (x)} \right|_{ - \infty }^{ + \infty } = 0, \\ \end{aligned}$$

(11.84)

where the property $\theta (x) \in C_{0}^{\gamma /2 - 1} [1 - \gamma ,\gamma - 1]$ is used. Similarly, for n = m, one can obtain

$$G_{n,m} (1) = ( - 1)^{n} n!\int {\theta (x)dx} = ( - 1)^{n} n!M_{0}^{\theta } = ( - 1)^{n} n!.$$

(11.85)

And for $n > m$, doing m times integrations by parts to Eq. (11.83), we have

$$\begin{aligned} G_{n,m} (1) & = \int {x^{n} \theta^{(m)} (x)dx} = x^{n} \theta^{(m - 1)} (x)|_{ - \infty }^{ + \infty } - n\int {x^{n - 1} \theta^{(m - 1)} (x)dx} \\ & = ( - 1)^{m} n!\int {x^{n - m} \theta (x)} dx/m! = ( - 1)^{n} n!M_{n - m}^{\theta } /m! = 0 \\ \end{aligned}$$

(11.86)

for $m = 0,1, \ldots ,\gamma /2 - 1$ and $n = m + 1,m + 2, \ldots ,\gamma - 1$. Eqs. (11.84)–(11.86) give the relation

$$G_{n,m} (1) = ( - 1)^{n} \delta_{n,m} n!.$$

(11.87)

By performing the same analysis shown in Eqs. (11.83)–(11.87), one can obtain $G_{n,m} ( - 1) = ( - 1)^{n + 1} \delta_{n,m} n!$ for $n = 0,1, \ldots ,\gamma - 1$ and $m = 0,1, \ldots ,\gamma /2 - 1$. From them, one can see that there is the relation $G_{n,m} (x) \equiv 0$ for $n \ne m$ since $G_{n,m} (1) = G_{n,m} ( - 1) = 0$ and $G_{n,m} (x) = 2^{m - n - 1} G_{n,m} (2x)$. Thus, we have

$$g_{n,m} (x) = \sum\limits_{{k \in {\mathbb{Z}}}} {(x - k)^{n} \theta^{(m)} (x - k)} = 0$$

(11.88)

for $n = 0,1, \ldots ,\gamma - 1$, $m = 0,1, \ldots ,\gamma /2 - 1$ and $n \ne m$.

When n = m, Eq. (11.82) gives $2G_{n,n} (x) = G_{n,n} (2x)$, i.e., $2\int_{0}^{x} {g_{n,n} (y)dy} = \int_{0}^{2x} {g_{n,n} (y)dy}$, which means the relation $g_{n,n} (x) = g_{n,n} (2x)$ for all x. Therefore, we have $g_{n,n} (x) = c$ which is a constant, since $g_{n,n} (x)$ is a continuous function. Finally, considering Eq. (11.87), one can obtain

$$g_{n,n} (x) \equiv g_{n,n} (1) = G_{n,n} (1) = ( - 1)^{n} n!.$$

(11.89)

According to Eqs. (11.88) and (11.89), the relation of Eq. (11.10) can be obtained.

Appendix 11.2 Multiresolution Decomposition of Interpolating Wavelet

Substituting the two-scale relation of Eq. (11.6) into the approximation Eq. (11.11) and using Eq. (11.7), we get

$$\begin{array}{*{20}l} {S^{j} f({\mathbf{x}}) = \sum\limits_{{k \in {\mathbb{Z}}}} {\sum\limits_{{l \in {\mathbb{Z}}}} {f({\mathbf{x}}_{j,k,l} )[\theta (2^{j + 1} x - 2k) + \sum\limits_{n \in Odd} {p_{n} \theta (2^{j + 1} x - 2k - n)} ]} } [\theta (2^{j + 1} x - 2l)} \hfill \\ {\quad \quad + \sum\limits_{m \in Odd} {p_{m} \theta (2^{j + 1} y - 2l - m)} ]} \hfill \\ { = \sum\limits_{k \in Even} {\sum\limits_{l \in Even} {f({\mathbf{x}}_{j + 1,k,l} )\theta_{j + 1,k,l} ({\mathbf{x}})} } + \sum\limits_{k \in Even} {\sum\limits_{l \in Odd} {[\sum\limits_{m \in Odd} {p_{m} f({\mathbf{x}}_{j + 1,k,l - m} )} ]\theta_{j + 1,k,l} ({\mathbf{x}})} } } \hfill \\ {\quad \quad + \sum\limits_{k \in Odd} {\sum\limits_{l \in Even} {[\sum\limits_{n \in Odd} {p_{n} f({\mathbf{x}}_{j + 1,k - n,l} )} ]\theta_{j + 1,k,l} ({\mathbf{x}})} } } \hfill \\ {\quad \quad + \sum\limits_{k \in Odd} {\sum\limits_{l \in Odd} {[\sum\limits_{n \in Odd} {\sum\limits_{m \in Odd} {p_{n} p_{m} f({\mathbf{x}}_{j + 1,k - n,l - m} )} } ]\theta_{j + 1,k,l} ({\mathbf{x}})} } } \hfill \\ \end{array} .$$

(11.90)

On the other hand, for $k \in Even$ and $l \in Odd$, one can obtain

$$\begin{aligned} & S^{j} f({\mathbf{x}}_{j + 1,k,l} ) = \sum\limits_{{n \in {\mathbb{Z}}}} {\sum\limits_{{m \in {\mathbb{Z}}}} {f({\mathbf{x}}_{j,n,m} )\theta (k/2 - n)\theta (l/2 - m)} } \\ & = \sum\limits_{{m \in {\mathbb{Z}}}} {f({\mathbf{x}}_{j + 1,j,k,m} )\sum\limits_{o \in Odd} {p_{o} \theta (l - 2m - o)} } = \sum\limits_{o \in Odd} {p_{o} f({\mathbf{x}}_{j + 1,k,l - o} )} , \\ \end{aligned}$$

(11.91)

where Eqs. (11.6) and (11.7) and the property $\theta (k) = \delta_{0,k}$ for $k \in {\mathbb{Z}}$ are considered. By using the same method, we can also obtain

$$\sum\limits_{o \in Odd} {p_{o} f({\mathbf{x}}_{j + 1,k - o,l} )} = S^{j} f({\mathbf{x}}_{j + 1,k,l} )\;{\text{for}}\;k \in Odd\;{\text{and}}\;l \in Even,$$

(11.92)

$$\sum\limits_{o \in Odd} {\sum\limits_{q \in Odd} {p_{o} p_{q} f({\mathbf{x}}_{j + 1,k - o,l - q} )} } = S^{j} f({\mathbf{x}}_{j + 1,k,l} )\;{\text{for}}\;k,l \in Odd.$$

(11.93)

Substitution of Eqs. (11.91)–(11.93) into (11.90) yields

$$S^{j} f({\mathbf{x}}) = \sum\limits_{k \in Even} {\sum\limits_{l \in Even} {f({\mathbf{x}}_{j + 1,k,l} )\theta_{j + 1,k,l} ({\mathbf{x}})} } + \sum\limits_{(k,l) \in \Re } {S^{j} f({\mathbf{x}}_{j + 1,k,l} )\theta_{j + 1,k,l} ({\mathbf{x}})}$$

(11.94)

in which $\Re$ is the set of all integer pair (k, l) with at least one element is an odd number.

Finally, Eq. (11.13) for $R^{j} f({\mathbf{x}})$ is obtained by subtracting $S^{j} f({\mathbf{x}})$ in Eq. (11.94) from $S^{j + 1} f({\mathbf{x}})$ defined by Eq. (11.11). Hence, the multiresolution decomposition of Eq. (11.12) is proved.

Appendix 11.3 Error Estimation of the Interpolating Wavelet Approximation Defined on the Whole Space

Here, we first examine the relation of Eq. (11.15). For any point x, one can always find a point ${\mathbf{x}}_{j,k,l}$ satisfying $\left| {{\mathbf{x}} - {\mathbf{x}}_{j,k,l} } \right| \le 2^{ - j} /\sqrt 2$. Then because both of $f({\mathbf{x}})$ and $S^{j} f({\mathbf{x}})$ are continuous functions, there exist two constants $\varepsilon_{0}$ and $\varepsilon_{1}$ satisfying

$$\left\| {f({\mathbf{x}}) - f({\mathbf{x}}_{j,k,l} )} \right\|_{\infty } \le \varepsilon_{0} \left| {{\mathbf{x}}- {\mathbf{x}}_{j,k,l} } \right| \le \varepsilon_{0} 2^{ - j} /\sqrt 2 ,$$

(11.95)

$$\left\| {S^{j} f({\mathbf{x}}) - S^{j} f({\mathbf{x}}_{j,k,l} )} \right\|_{\infty } \le \varepsilon_{1} \left| {{\mathbf{x}} - {\mathbf{x}}_{j,k,l} } \right| \le \varepsilon_{1} 2^{ - j} /\sqrt 2 .$$

(11.96)

Then, one can obtain

$$\begin{aligned} & \left\| {f({\mathbf{x}}) - S^{j} f({\mathbf{x}})} \right\|_{\infty } = \left\| {[f({\mathbf{x}}) - f({\mathbf{x}}_{j,k,l} )] + [S^{j} f({\mathbf{x}}_{j,k,l} ) - S^{j} f({\mathbf{x}})]} \right\|_{\infty } \\ & \quad \le \left\| {f({\mathbf{x}}) - f({\mathbf{x}}_{j,k,l} )} \right\|_{\infty } + \left\| {S^{j} f({\mathbf{x}}_{j,k,l} ) - S^{j} f({\mathbf{x}})} \right\|_{\infty } \le C_{0,\infty } 2^{j} , \\ \end{aligned}$$

(11.97)

which gives the relation of Eq. (11.15).

For the relation of Eq. (11.16), by using the Taylor expansion for $f({\mathbf{x}})$ at the point x, we have

$$f({\tilde{\mathbf{x}}}) = \sum\limits_{n = 0}^{\lambda - 1} {\frac{1}{n!}[(\tilde{x} - x)\frac{\partial }{\partial x} + (\tilde{y} - y)\frac{\partial }{\partial y}]^{n} f({\mathbf{x}})} + \frac{1}{\lambda !}[(\tilde{x} - x)\frac{\partial }{\partial x} + (\tilde{y} - y)\frac{\partial }{\partial y}]^{\lambda } f({\mathbf{x}}_{\theta } ),$$

(11.98)

where ${\mathbf{x}}_{\theta }$ is on the rectangle with two diagonal apexes ${\tilde{\mathbf{x}}}$ and x, and

$$[(\tilde{x} - x)\frac{\partial }{\partial x} + (\tilde{y} - y)\frac{\partial }{\partial y}]^{n} f({\mathbf{x}}) = \sum\limits_{m = 0}^{n} {C_{n}^{m} (\tilde{x} - x)^{m} (\tilde{y} - y)^{n - m} D_{n}^{m} f({\mathbf{x}})}$$

(11.99)

with $D_{n}^{m} f({\mathbf{x}}) = \left. {\frac{{\partial^{n} f}}{{\partial x^{m} \partial y^{n - m} }}} \right|_{{\mathbf{x}}}$. Assigning ${\tilde{\mathbf{x}}} = {\mathbf{x}}_{j,k,l}$ to Eq. (11.98) and then substituting them into Eq. (11.11), one gains

$$\begin{aligned} S^{j} f({\mathbf{x}}) & = \sum\limits_{{k \in {\mathbb{Z}}}} {\sum\limits_{{ \in {\mathbb{Z}}}} {\sum\limits_{n = 0}^{\lambda - 1} {\frac{1}{n!}\sum\limits_{m = 0}^{n} {C_{n}^{m} (\frac{k}{{2^{j} }} - x)^{m} (\frac{l}{{2^{j} }} - y)^{n - m} D_{n}^{m} f({\mathbf{x}})\theta_{j,k,l} ({\mathbf{x}})} } } } \\ & \quad + \sum\limits_{{k \in {\mathbb{Z}}}} {\sum\limits_{{l \in {\mathbb{Z}}}} {\frac{1}{\lambda !}\sum\limits_{m = 0}^{\lambda } {C_{n}^{m} (\frac{k}{{2^{j} }} - x)^{m} (\frac{l}{{2^{j} }} - y)^{\lambda - m} D_{\lambda }^{m} f({\mathbf{x}}_{\theta ,k,l} )} } } \\ & = \sum\limits_{n = 0}^{\lambda - 1} {\frac{{2^{ - jn} }}{n!}\sum\limits_{m = 0}^{n} {[C_{n}^{m} \sum\limits_{{k \in {\mathbb{Z}}}} {(k - 2^{j} x)^{m} \theta (2^{j} x - k)} \sum\limits_{{l \in {\mathbb{Z}}}} {(l - 2^{j} y)^{n - m} \theta (2^{j} y - l)} D_{n}^{m} f({\mathbf{x}})]} } \\ & \quad + \frac{{2^{ - j\lambda } }}{\lambda !}\sum\limits_{m = 0}^{\lambda } {[C_{\lambda }^{m} \sum\limits_{{k \in {\mathbb{Z}}}} {\sum\limits_{{l \in {\mathbb{Z}}}} {(k - 2^{j} x)^{m} (l - 2^{j} y)^{\lambda - m} \theta_{j,k,l} ({\mathbf{x}})D_{\lambda }^{m} f({\mathbf{x}}_{\theta ,k,l} )} } ]} \\ \end{aligned}$$

(11.100)

in which ${\mathbf{x}}_{\theta ,k,l}$ is on the rectangle with two diagonal apexes ${\mathbf{x}}_{j,k,l}$ and x. Then, following the property of Eq. (11.10), we have

$$\sum\limits_{{k \in {\mathbb{Z}}}} {(k - 2^{j} x)^{q} \theta (2^{j} x - k)} = \delta_{0,q} \;{\text{for}}\;q = 0,1, \ldots ,\gamma - 1.$$

(11.101)

Substituting Eqs. (11.101) into (11.100), and considering $\lambda \le \gamma$, we have

$$S^{j} f({\mathbf{x}}) = f({\mathbf{x}}) + \frac{{2^{ - j\lambda } }}{\lambda !}\sum\limits_{m = 0}^{\lambda } {[C_{\lambda }^{m} \sum\limits_{{k \in {\mathbb{Z}}}} {\sum\limits_{{l \in {\mathbb{Z}}}} {(k - 2^{j} x)^{m} (l - 2^{j} y)^{\lambda - m} \theta_{j,k,l} ({\mathbf{x}})D_{\lambda }^{m} f({\mathbf{x}}_{\theta ,k,l} )} } ]} .$$

(11.102)

Since $\Omega_{j,k,l}^{\inf } = [2^{ - j} (1 - \gamma + k),2^{ - j} (\gamma + k - 1)] \times [2^{ - j} (1 - \gamma + l),2^{ - j} (\gamma + l - 1)]$ and $\theta_{j,k,l} ({\mathbf{x}}) \in C_{0}^{\gamma /2 - 1} \Omega_{j,k,l}^{\inf }$, we have the relations

$$\left\| {(k - 2^{j} x)^{m} (l - 2^{j} y)^{\lambda - m} \theta_{j,k,l} ({\mathbf{x}})} \right\|_{\infty } \le (\gamma - 1)^{\lambda } \left\| {\theta_{j,k,l} ({\mathbf{x}})} \right\|_{\infty } \;{\text{for all}}\;k,l \in {\mathbb{Z}},$$

(11.103)

and for any point x

$$\theta_{j,k,l} ({\mathbf{x}}) = 0\;{\text{when}}\;k \notin (2^{j} x - \lambda + 1,2^{j} x + \lambda - 1)\;{\text{or}}\;l \notin (2^{j} y - \lambda + 1,2^{j} y + \lambda - 1).$$

(11.104)

Finally, following Eqs. (11.100)–(11.104), one can obtain

$$\begin{aligned} & \left\| {f({\mathbf{x}}) - S^{j} f({\mathbf{x}})} \right\|_{\infty } \le 2^{ - j\lambda } \frac{{(2\gamma - 1)^{2} (\gamma - 1)^{\lambda } }}{\lambda !}\left\| {\theta_{j,k,l} ({\mathbf{x}})} \right\|_{\infty } \sum\limits_{m = 0}^{\lambda } {C_{\lambda }^{m} \left\| {D_{\lambda }^{m} f({\mathbf{x}}_{\theta ,k,l} )} \right\|_{\infty } } \\ & \quad \le C_{1,\infty } 2^{ - j\lambda } \\ \end{aligned}$$

(11.105)

in which the properties $\theta_{j,k,l} ({\mathbf{x}}) \in C^{0} (R^{2} )$ and $f({\mathbf{x}}) \in C^{\lambda } (R^{2} )$ are considered. Specially, if $D_{\lambda }^{m} f({\mathbf{x}}) \equiv 0$ for $m = 0,1, \ldots ,\lambda$, we have $C_{1,\infty } = 0$ from Eq. (11.105). On the basis of Eq. (11.105), it is very easy to check Eq. (10.17) holding.

Appendix 11.4 Error Estimation of the Interpolating Wavelet Approximation Defined on a Finite Domain

Since $f({\mathbf{x}}) \in C^{\mu } (\Omega )$, one can always find a way to supplement the definition of function $f({\mathbf{x}})$ in the domain $\overline{\Omega }$ satisfying $f({\mathbf{x}}) \in C^{\mu } (\Omega \cup \overline{\Omega })$. Then following Eqs. (11.20)–(11.22), we have

$$\begin{array}{*{20}l} {S_{L}^{j} f({\mathbf{x}}) = \sum\limits_{l = 1}^{{N_{j} }} {f({\mathbf{x}}_{l} )\theta_{{j,kx({\mathbf{x}}_{l} ),ky({\mathbf{x}}_{l} )}} ({\mathbf{x}})} + \sum\limits_{l = 1}^{{N_{{\overline{\Omega }}} }} {f({\overline{\mathbf{x}}}_{l} )\theta_{{j,kx({\overline{\mathbf{x}}}_{l} ),ky({\overline{\mathbf{x}}}_{l} )}} ({\mathbf{x}})} } \hfill \\ { + \sum\limits_{l = 1}^{{N_{{\overline{\Omega }}} }} {[\tilde{f}_{l} ({\overline{\mathbf{x}}}_{l} ) - f({\overline{\mathbf{x}}}_{l} )]\theta_{{j,kx({\overline{\mathbf{x}}}_{l} ),ky({\overline{\mathbf{x}}}_{l} )}} ({\mathbf{x}})} } \hfill \\ { = S^{j} f({\mathbf{x}}) + \sum\limits_{l = 1}^{{N_{{\overline{\Omega }}} }} {[\tilde{f}_{l} ({\overline{\mathbf{x}}}_{l} ) - f({\overline{\mathbf{x}}}_{l} )]\theta_{{j,kx({\overline{\mathbf{x}}}_{l} ),ky({\overline{\mathbf{x}}}_{l} )}} ({\mathbf{x}})} } \hfill \\ \end{array} \;{\text{for}}\;{\mathbf{x}} \in \Omega .$$

(11.106)

By applying Theorem 11.1 and Eq. (11.106), one can obtain

$$\begin{aligned} & \left\| {S_{L}^{j} f({\mathbf{x}}) - f({\mathbf{x}})} \right\|_{\infty } = \left\| {S^{j} f({\mathbf{x}}) - f({\mathbf{x}})} \right\|_{\infty } + \sum\limits_{l = 1}^{{N_{{\overline{\Omega }}} }} {\left\| {[\tilde{f}_{l} ({\overline{\mathbf{x}}}_{l} ) - f({\overline{\mathbf{x}}}_{l} )]\theta_{{j,kx({\overline{\mathbf{x}}}_{l} ),ky({\overline{\mathbf{x}}}_{l} )}} ({\mathbf{x}})} \right\|_{\infty } } \\ & \quad \le C_{1,\infty } 2^{ - j\lambda } + \sum\limits_{l = 1}^{{N_{{\overline{\Omega }}} }} {\left\| {[\tilde{f}_{l} ({\overline{\mathbf{x}}}_{l} ) - f({\overline{\mathbf{x}}}_{l} )]\theta_{{j,kx({\overline{\mathbf{x}}}_{l} ),ky({\overline{\mathbf{x}}}_{l} )}} ({\mathbf{x}})} \right\|_{\infty } } \\ \end{aligned}$$

(11.107)

for ${\mathbf{x}} \in \Omega$. Since $\tilde{f}_{l} ({\mathbf{x}})$ is a Lagrange interpolation with order $\eta_{l}$ of function $f({\mathbf{x}})$, by using the relations $\left| {\overline{x}_{l} - \tilde{x}_{l,i} } \right| \le \alpha_{1} 2^{ - j}$ and $\left| {\overline{y}_{l} - \tilde{y}_{l,i} } \right| \le \alpha_{2} 2^{ - j}$, we get [16]

$$\left| {f({\overline{\mathbf{x}}}_{l} ) - \tilde{f}_{l} ({\overline{\mathbf{x}}}_{l} )} \right| \le \varepsilon_{2} 2^{{ - j\min \left\{ {\mu ,\eta_{l} } \right\}}} ,$$

(11.108)

where $\varepsilon_{2}$ is a constant with property $\varepsilon_{2} = 0$ when $D_{\lambda }^{m} f({\mathbf{x}}) \equiv 0$ for $m = 0,1, \ldots ,\lambda$. Finally, by considering Eq. (11.104) and $\theta_{j,k,l} ({\mathbf{x}}) \in C_{0}^{\gamma /2 - 1} \Omega_{j,k,l}^{\inf }$, Eq. (11.107) can be expressed as

$$\left\| {S_{L}^{j} f({\mathbf{x}}) - f({\mathbf{x}})} \right\|_{\infty } \le [C_{1,\infty } + (2\gamma - 1)^{2} \varepsilon_{4} ]2^{ - j\lambda } = C_{2,\infty } 2^{ - j\lambda } \;{\text{for}}\;{\mathbf{x}} \in \Omega ,$$

(11.109)

where $C_{2,\infty } = 0$ when $D_{\lambda }^{m} f({\mathbf{x}}) \equiv 0$ for $m = 0,1, \ldots ,\lambda$. Based on Eq. (11.109), finally, there is no difficulty for one to check Eq. (11.25) holding.

Appendix 11.5 Proof of the Interpolating Property for the Modified Multiresolution Approximation

If J = j₀ in Eq. (11.28) , following the delta function property of the modified wavelet basis $\phi_{n} ({\mathbf{x}}_{k} )$ discussed after Eq. (11.23), we can directly obtain

$$P_{{j_{0} }}^{{j_{0} }} f({\mathbf{x}}_{k} ) \equiv S^{{j_{0} }} f({\mathbf{x}}_{k} ) = f({\mathbf{x}}_{k} )\;{\text{for all nodes}}\;{\mathbf{x}}_{k} .$$

(11.110)

Then we assume

$$P_{{j_{0} }}^{o} f({\mathbf{x}}_{j,n,m} ) \equiv f({\mathbf{x}}_{j,n,m} )\;{\text{for}}\;(n,m) \in \Re_{j} \;{\text{and}}\;j_{0} \le j \le o.$$

(11.111)

Following Eqs. (11.27) and (11.29), we obtain

$$P_{{j_{0} }}^{o + 1} f({\mathbf{x}}_{j,n,m} ) = P_{{j_{0} }}^{o} f({\mathbf{x}}_{j,n,m} ) + \sum\limits_{{(k,l) \in \Re_{o} }} {[f({\mathbf{x}}_{o + 1,k,l} ) - P_{{j_{0} }}^{o} f({\mathbf{x}}_{o + 1,k,l} )]\theta_{o + 1,k,l} } ({\mathbf{x}}_{j,n,m} )$$

(11.112)

for $(n,m) \in \Re_{j}$ and $j_{0} \le j \le o + 1$.

For all nodes ${\mathbf{x}}_{j,n,m}$, $j_{0} \le j \le o$, one can gain

$$\theta_{o + 1,k,l} ({\mathbf{x}}_{j,n,m} ) = \theta (2^{o + 1 - j} n - k)\theta (2^{o + 1 - j} m - l) = 0$$

by considering the interpolation property $\theta (q) = \delta_{0,q}$ for all $q \in {\mathbb{Z}}$. Because both $2^{o + 1 - j} n$ and $2^{o + 1 - j} m$ are even numbers, and at least one of k and l is odd number, therefore, Eq. (11.112) and the assumption (11.111) give the following equation

$$P_{{j_{0} }}^{o + 1} f({\mathbf{x}}_{j,n,m} ) = P_{{j_{0} }}^{o} f({\mathbf{x}}_{j,n,m} ) = f({\mathbf{x}}_{j,n,m} )\;{\text{for}}\;j_{0} \le j \le o.$$

(11.113)

For $j = o + 1$ in Eq. (11.112), by using the interpolation property $\theta_{o + 1,k,l} ({\mathbf{x}}_{o + 1,n,m} ) = \theta (n - k)\theta (m - l) = \delta_{n,k} \delta_{m,l}$, Eq. (11.112) can be reduced into

$$\begin{aligned} & P_{{j_{0} }}^{o + 1} f({\mathbf{x}}_{o + 1,n,m} ) = P_{{j_{0} }}^{o} f({\mathbf{x}}_{o + 1,n,m} ) + \sum\limits_{{(k,l) \in \Re_{o} }} {[f({\mathbf{x}}_{o + 1,k,l} ) - P_{{j_{0} }}^{o} f({\mathbf{x}}_{o + 1,k,l} )]\delta_{n,k} \delta_{m,l} } \\ & \quad = P_{{j_{0} }}^{o} f({\mathbf{x}}_{o + 1,n,m} ) + f({\mathbf{x}}_{o + 1,n,m} ) - P_{{j_{0} }}^{o} f({\mathbf{x}}_{o + 1,n,m} ) = f({\mathbf{x}}_{o + 1,n,m} ) \\ \end{aligned}$$

(11.114)

for $j = o + 1$. Hence, Eqs. (11.113) and (11.114) give

$$P_{{j_{0} }}^{o + 1} f({\mathbf{x}}_{j,n,m} ) \equiv f({\mathbf{x}}_{j,n,m} )\;{\text{for}}\;(n,m) \in \Re_{j} \;{\text{and}}\;j_{0} \le j \le o + 1.$$

(11.115)

Finally, combining the recurrence relations (11.111) and (11.115) with the starting condition of Eq. (11.110), the Proposition 11.2 is proved.

Appendix 11.6 Error Estimation of the Modified Interpolating Multiresolution Approximation

Following the relations of Eqs. (11.26) and (11.27), one can obtain

$$\begin{aligned} & \left\| {f({\mathbf{x}}) - P_{{j_{0} }}^{n} f({\mathbf{x}})} \right\|_{\infty } \le \left\| {f({\mathbf{x}}) - P_{{j_{0} }}^{n - 1} f({\mathbf{x}})} \right\|_{\infty } \\ & \quad + \sum\limits_{{(k,l) \in \Re_{n - 1} }} {\left| {f({\mathbf{x}}_{j + 1,k,l} ) - P_{{j_{0} }}^{n - 1} f({\mathbf{x}}_{j + 1,k,l} )} \right| \times \left\| {\theta_{j + 1,k,l} ({\mathbf{x}})} \right\|_{\infty } } \\ \end{aligned}$$

(11.116)

for $n > j_{0}$.

By considering $\theta_{j,k,l} ({\mathbf{x}}) \in C_{0}^{\gamma /2 - 1} \Omega_{j,k,l}^{\inf }$ and Eqs. (11.104), (11.116) can be expressed as

$$\left\| {f({\mathbf{x}}) - P_{{j_{0} }}^{n} f({\mathbf{x}})} \right\|_{\infty } \le \varepsilon_{3} \left\| {f({\mathbf{x}}) - P_{{j_{0} }}^{n - 1} f({\mathbf{x}})} \right\|_{\infty }$$

(11.117)

in which $\varepsilon_{3}$ is a constant. By applying Eq. (11.117) iteratively, and considering the relation $P_{{j_{0} }}^{{j_{0} }} f({\mathbf{x}}) = S_{L}^{{j_{0} }} f({\mathbf{x}})$ and Theorem 11.2, one can obtain

$$\left\| {f({\mathbf{x}}) - P_{{j_{0} }}^{J} f({\mathbf{x}})} \right\|_{\infty } \le \varepsilon_{4} C_{2,\infty } \left\| {f({\mathbf{x}}) - S_{L}^{{j_{0} }} f({\mathbf{x}})} \right\|_{\infty } \le C_{3,\infty } 2^{{ - j_{0} \lambda }} ,$$

(11.118)

where $\varepsilon_{4}$ is a constant, and $C_{3,\infty } = 0$ when $D_{\lambda }^{m} f({\mathbf{x}}) \equiv 0$ for $m = 0,1, \ldots ,\lambda$. On the basis of Eq. (11.118), there is no difficulty for one to check Eq. (11.30) holding.

Appendix 11.7 Construction of the Targeted Interpolation Based on Interpolating Wavelet

The method of mathematic induction is employed here to prove the expression of Eq. (11.34). First, for $P_{{j_{0} }}^{{j_{0} }} f({\mathbf{x}})$ one can directly obtain the relation

$$P_{{j_{0} }}^{{j_{0} }} f({\mathbf{x}}) = \sum\limits_{{n \in \Re_{{j_{0} }} }} {f({\mathbf{x}}_{n} )\phi_{n} ({\mathbf{x}})} = \sum\limits_{{n \in \Re_{{j_{0} }} }} {a_{n} \phi_{n} ({\mathbf{x}})} = \sum\limits_{{n \in \Re_{{j_{0} }}^{I} }} {f({\mathbf{x}}_{n} )\Theta_{{j_{0} ,n}} ({\mathbf{x}})} + \sum\limits_{{n \in \Re_{{j_{0} }}^{R} }} {c_{n} \Theta_{{j_{0} ,n}} ({\mathbf{x}})} ,$$

(11.119)

where the basis function $\Theta_{{j_{0} ,n}} ({\mathbf{x}}) = \phi_{n} ({\mathbf{x}})$ for $n \in \Re_{{j_{0} }} = \Re_{{j_{0} }}^{I} \cup \Re_{{j_{0} }}^{R}$. And it is also easy to check the relation $P_{{j_{0} }}^{{j_{0} }} f({\mathbf{x}}_{n} ) \equiv f({\mathbf{x}}_{n} )$ held for $n \in \Re_{{j_{0} }}^{I}$ by using the property $\phi_{n} ({\mathbf{x}}_{m} ) = \delta_{n,m}$ for $\rho (m) \le \rho (n)$. Based on such a property, in fact, we can further directly obtain the relation $P_{{j_{0} }}^{{j_{\max } }} f({\mathbf{x}}_{n} ) \equiv f({\mathbf{x}}_{n} )$ for $n \in \Re_{{j_{0} }}$.

Next, we assume that there is the expression

$$P_{{j_{0} }}^{j} f({\mathbf{x}}) = \sum\limits_{{n \in \Re_{ \le j}^{I} }} {f({\mathbf{x}}_{n} )\Theta_{j,n} ({\mathbf{x}})} + \sum\limits_{{n \in \Re_{ \le j}^{R} }} {c_{n} \Theta_{j,n} ({\mathbf{x}})} = \sum\limits_{{n \in \Re_{ \le j} }} {a_{n} \Theta_{j,n} ({\mathbf{x}})}$$

(11.120)

in which $\Re_{ \le j}^{I} = \cup_{{q = j_{0} }}^{j} \Re_{q}^{I}$ and $\Re_{ \le j}^{R} = \cup_{{q = j_{0} }}^{j} \Re_{q}^{R}$. Substitution of Eqs. (11.120) into  (11.34) leads to

$$\begin{aligned} P_{{j_{0} }}^{j + 1} f({\mathbf{x}}) & = P_{{j_{0} }}^{j} f({\mathbf{x}}) + \sum\limits_{{n \in \Re_{j + 1}^{I} }} {[f({\mathbf{x}}_{n} ) - P_{{j_{0} }}^{j} f({\mathbf{x}}_{n} )]\phi_{n} ({\mathbf{x}})} + \sum\limits_{{n \in \Re_{j + 1}^{R} }} {c_{n} \phi_{n} ({\mathbf{x}})} \\ & = \sum\limits_{{n \in \Re_{ \le j}^{I} }} {f({\mathbf{x}}_{n} )\Theta_{j,n} ({\mathbf{x}})} + \sum\limits_{{n \in \Re_{j + 1}^{I} }} {f({\mathbf{x}}_{n} )\phi_{n} ({\mathbf{x}})} + \sum\limits_{{n \in \Re_{ \le j}^{R} }} {c_{n} \Theta_{j,n} ({\mathbf{x}})} + \sum\limits_{{n \in \Re_{j + 1}^{R} }} {c_{n} \phi_{n} ({\mathbf{x}})} \\ & \quad - \sum\limits_{{n \in \Re_{j + 1}^{I} }} {\sum\limits_{{m \in \Re_{ \le j}^{I} }} {f({\mathbf{x}}_{m} )\Theta_{j,m} ({\mathbf{x}}_{n} )\phi_{n} ({\mathbf{x}})} } - \sum\limits_{{n \in \Re_{j + 1}^{I} }} {\sum\limits_{{m \in \Re_{ \le j}^{R} }} {c_{m} \Theta_{j,m} ({\mathbf{x}}_{n} )\phi_{n} ({\mathbf{x}})} } \\ & = \sum\limits_{{n \in \Re_{ \le j}^{I} }} {f({\mathbf{x}}_{n} )[\Theta_{j,n} ({\mathbf{x}}) - \sum\limits_{{m \in \Re_{j + 1}^{I} }} {\Theta_{j,n} ({\mathbf{x}}_{m} )\phi_{m} ({\mathbf{x}})} ]} + \sum\limits_{{n \in \Re_{j + 1}^{I} }} {f({\mathbf{x}}_{n} )\phi_{n} ({\mathbf{x}})} \\ & \quad + \sum\limits_{{n \in \Re_{ \le j}^{R} }} {c_{n} [\Theta_{j,n} ({\mathbf{x}}) - \sum\limits_{{m \in \Re_{j + 1}^{I} }} {\Theta_{j,n} ({\mathbf{x}}_{m} )\phi_{m} ({\mathbf{x}})} ]} + \sum\limits_{{n \in \Re_{j + 1}^{R} }} {c_{n} \phi_{n} ({\mathbf{x}})} \\ & = \sum\limits_{{n \in \Re_{ \le j + 1}^{I} }} {f({\mathbf{x}}_{n} )\Theta_{j + 1,n} ({\mathbf{x}})} + \sum\limits_{{n \in \Re_{ \le j + 1}^{R} }} {c_{n} \Theta_{j + 1,n} ({\mathbf{x}})} = \sum\limits_{{n \in \Re_{ \le j + 1} }} {a_{n} \Theta_{j + 1,n} ({\mathbf{x}})} . \\ \end{aligned}$$

(11.121)

By iteratively using the relation of Eq. (11.121) and the starting condition (11.119), one can obtain the formula of Eq. (11.34).

It can be seen from Eqs. (11.119)–(11.121) that the function $\Theta_{j,n} ({\mathbf{x}})$ for $j_{0} \le j \le j_{\max }$ and $n \in \Re$ is a linear combination of modified wavelet basis functions $\phi_{n} ({\mathbf{x}})$. Therefore, Eq. (11.120) can be rewritten into the matrix form

$$P_{{j_{0} }}^{j} f({\mathbf{x}}) = {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{B}}^{j} {\mathbf{a}}$$

(11.122)

in which the vectors ${{\varvec{\Phi}}}({\mathbf{x}}) = \left\{ {\phi_{n} ({\mathbf{x}}),n = 1,2, \ldots ,N} \right\}$ and ${\mathbf{a}} = \left\{ {a_{n} ,n = 1,2, \ldots ,N} \right\}^{{\text{T}}}$, and ${\mathbf{B}}^{j}$ is the N × N transformation matrix. Based on Eq. (11.119), one can directly obtain

$${\mathbf{B}}^{{j_{0} }} = \left\{ {b_{kk}^{{j_{0} }} = 1\;{\text{for}}\;k = \Re_{{j_{0} }} ,\;b_{kl}^{{j_{0} }} = 0\;{\text{otherwise}}} \right\}.$$

(11.123)

Substituting Eqs. (11.122) into (11.121), we get

$$P_{{j_{0} }}^{j + 1} f({\mathbf{x}}) = {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{B}}^{j} {\mathbf{a}} + {{\varvec{\Phi}}}({\mathbf{x}})\overline{{\mathbf{B}}}^{j + 1} {\mathbf{a}} - {{\varvec{\Phi}}}({\mathbf{x}})\widetilde{{\mathbf{B}}}^{j + 1} \overline{{{\varvec{\Phi}}}}^{j + 1} {\mathbf{B}}^{j} {\mathbf{a}} = {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{B}}^{j + 1} {\mathbf{a}}$$

(11.124)

in which

$$\overline{{\mathbf{B}}}^{j + 1} = \left\{ {\overline{b}_{kk}^{j + 1} = 1\;{\text{for}}\;k = \Re_{j + 1} ,\;\overline{b}_{kl}^{j + 1} = 0\;{\text{otherwise}}} \right\},$$

(11.125)

$$\widetilde{{\mathbf{B}}}^{j + 1} = \left\{ {\tilde{b}_{kk}^{j + 1} = 1\;{\text{for}}\;k = \Re_{j + 1}^{I} ,\tilde{b}_{kl}^{j + 1} = 0\;{\text{otherwise}}} \right\},$$

(11.126)

$$\overline{{{\varvec{\Phi}}}}^{j + 1} = \left\{ {\overline{\Phi }_{kl}^{j + 1} = \phi_{l} ({\mathbf{x}}_{k} )\;{\text{for}}\;k \in \Re_{j + 1}^{I} \, ,\overline{\Phi }_{kl}^{j + 1} = 0\;{\text{otherwise}}} \right\}.$$

(11.127)

By considering the above sparse features of $\widetilde{{\mathbf{B}}}^{j + 1}$ and $\overline{{{\varvec{\Phi}}}}^{j + 1}$, it is easy to check the relation $\overline{{{\varvec{\Phi}}}}^{j + 1} = \widetilde{{\mathbf{B}}}^{j + 1} \overline{{{\varvec{\Phi}}}}^{j + 1}$. Thus, Eq. (11.124) gives

$${\mathbf{B}}^{j + 1} = {\mathbf{B}}^{j} + \overline{{\mathbf{B}}}^{j + 1} - \overline{{{\varvec{\Phi}}}}^{j + 1} {\mathbf{B}}^{j} .$$

(11.128)

Considering that the wavelet basis function $\phi_{n} ({\mathbf{x}})$ has the compact support domain $\Omega_{n}$ and the function $P_{{j_{0} }}^{j} f({\mathbf{x}})$ defined by Eq. (11.120) is completely independent of $\phi_{n} ({\mathbf{x}})$ with $\rho (n) > j$, there is the property $\overline{\Phi }_{kl}^{j + 1} = 0$ for $\, k \notin \Re_{j + 1}^{I}$ or $l \notin \Re_{ \le j}^{T,k}$ for the matrix $\overline{{{\varvec{\Phi}}}}^{j + 1}$ with $\Re_{ \le j}^{T,k} = \Re_{ \le j}^{T} \cap \Re_{k}^{\Omega }$. Here the nodal set $\left\{ {{\mathbf{x}}_{n} ,n \in \Re_{ \le j}^{T} } \right\}$ is consisted of all those nodes with $\rho (n) \le j$, whose compact support domain $\Omega_{n}$ contains at least one node ${\mathbf{x}}_{m}$ for $\, m \in \Re^{I}$. And the nodal set $\left\{ {{\mathbf{x}}_{m} ,m \in \Re_{k}^{\Omega } } \right\}$ is consisted of all nodes whose compact support domain $\Omega_{m}$ contains the node ${\mathbf{x}}_{k}$.

By using the above property of $\overline{{{\varvec{\Phi}}}}^{j}$, the transformation matrix ${\mathbf{B}}^{j}$ can be expressed as

$${\mathbf{B}}^{j} = {\mathbf{B}}_{d}^{j} + {\mathbf{B}}_{s}^{j}$$

(11.129)

in which the incomplete identity matrix ${\mathbf{B}}_{d}^{j} = \left\{ {b_{d,kk}^{j} = 1\;{\text{for}}\;k \in \Re_{ \le j} ,\;b_{d,kl}^{j} = 0\;{\text{otherwise}}} \right\}$ and the sparse matrix ${\mathbf{B}}_{s}^{j}$ are verified for all elements $b_{s,kl}^{j} = 0$ for $k \notin \Re^{I}$, $l \notin \Re_{ \le j - 1}^{T,k}$ or $k = l$.

Based on the definition in Eq. (11.123) of ${\mathbf{B}}^{{j_{0} }}$, it is very easy to obtain ${\mathbf{B}}_{d}^{{j_{0} }} = {\mathbf{B}}^{{j_{0} }}$ and ${\mathbf{B}}_{s}^{{j_{0} }} = {\mathbf{0}}$. Then, substituting Eqs. (11.129) into (11.128), we gain

$${\mathbf{B}}^{j + 1} = ({\mathbf{B}}_{d}^{j} + \overline{{\mathbf{B}}}^{j + 1} ) + {\mathbf{B}}_{s}^{j} - \overline{{{\varvec{\Phi}}}}^{j + 1} B_{d}^{j} - \overline{{{\varvec{\Phi}}}}^{j + 1} {\mathbf{B}}_{s}^{j} = {\mathbf{B}}_{d}^{j + 1} + {\mathbf{B}}_{s}^{j + 1} .$$

(11.130)

By directly examining the definition of $\overline{{\mathbf{B}}}^{j + 1}$ given in Eq. (11.125), it is easy to check that the matrix ${\mathbf{B}}_{d}^{j + 1} = {\mathbf{B}}_{d}^{j} + \overline{{\mathbf{B}}}^{j + 1}$ is exactly the incomplete identity matrix defined in Eq. (11.129). By applying the properties $\overline{\Phi }_{kl}^{j + 1} = 0$ for $\, k \notin \Re_{j + 1}^{I}$ or $l \notin \Re_{ \le j}^{T,k}$, $b_{s,kl}^{j} = 0$ for $k \notin \Re^{I}$ or $l \notin \Re_{ \le j - 1}^{T,k}$, and the definition of ${\mathbf{B}}_{d}^{j}$ given in Eqs. (11.129), (11.130) gives

$${\mathbf{B}}_{s}^{j + 1} = {\mathbf{B}}_{s}^{j} - \overline{{{\varvec{\Phi}}}}^{j + 1} - \overline{{{\varvec{\Phi}}}}^{j + 1} {\mathbf{B}}_{s}^{j}$$

(11.131)

which verifies the sparsity pattern $b_{s,kl}^{j + 1} = 0$ for $k \notin \Re^{I}$, $l \notin \Re_{ \le j - 1}^{T,k}$ or $k = l$.

In Eq. (11.131), the matrix $\overline{{{\varvec{\Phi}}}}^{j + 1}$ has been redefined as

$$\overline{{{\varvec{\Phi}}}}^{j + 1} = \left\{ {\overline{\Phi }_{kl}^{j + 1} = \phi_{l} ({\mathbf{x}}_{k} )\;{\text{for}}\;k \in \Re_{j + 1}^{I} \;{\text{and}}\;l \in \Re_{ \le j}^{T} ,\overline{\Phi }_{kl}^{j + 1} = 0\;{\text{otherwise}}} \right\},$$

(11.132)

where the properties $\overline{\Phi }_{kl}^{j + 1} = 0$ for $l \notin \Re_{ \le j}^{T}$ is applied.

In addition, since $b_{s,kl}^{j + 1} = 0$ for $k \notin \Re^{I}$, there is the relation $\overline{{{\varvec{\Phi}}}}^{j + 1} {\mathbf{B}}_{s}^{j} = {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi } }}^{j + 1} {\mathbf{B}}_{s}^{j}$ with the sparse matrix ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi } }}^{j + 1} = \left\{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi }_{kl}^{j + 1} = \phi_{l} ({\mathbf{x}}_{k} )\;{\text{for}}\;k \in \Re_{j + 1}^{I} \;{\text{and}}\;l \in \Re_{ \le j}^{I} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi }_{kl}^{j + 1} = 0\;{\text{otherwise}}} \right\}$.

Therefore, the final transformation matrix can be expressed as ${\mathbf{B}}^{{j_{\max } }} = {\mathbf{I}} + {\mathbf{B}}_{s}^{{J_{I} }}$ with the identity matrix I and the sparse matrix ${\mathbf{B}}_{s}^{j}$ satisfying the recurrence relation

$${\mathbf{B}}_{s}^{j + 1} = {\mathbf{B}}_{s}^{j} - \overline{{{\varvec{\Phi}}}}^{j + 1} - {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi } }}^{j + 1} {\mathbf{B}}_{s}^{j} \;{\text{with}}\;{\mathbf{B}}_{s}^{{j_{0} }} = {\mathbf{0}},$$

(11.133)

where the property $\overline{{{\varvec{\Phi}}}}^{j} = {\mathbf{0}}$ for $j > J_{I}$ with $J_{I} = \max \left\{ {\rho (n),n \in \Re^{I} } \right\}$ has been considered.

By directly reducing the dimensions of matrices ${\mathbf{B}}_{s}^{j}$, and according to the sparsity features $\overline{{{\varvec{\Phi}}}}^{j}$ and ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi } }}^{j}$ in Eq. (11.133) displayed above, finally, one can obtain the shape functions $\left\{ {\Theta_{{j_{\max } ,n}} ({\mathbf{x}}), \, n \in \Re } \right\} = {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{B}}^{{j_{\max } }}$ defined by Eq. (11.35).