A wavelet multiresolution interpolation Galerkin method for targeted local solution enrichment

Abstract

A novel wavelet multiresolution interpolation formula is developed for approximating continuous functions defined on an arbitrary two-dimensional domain represented by a set of scattered nodes. The present wavelet interpolant is created explicitly without the need for matrix inversion. It possesses the Kronecker delta function property and does not contain any ad-hoc parameters, leading to an excellent stability and usefulness for function approximation. Using the wavelet multiresolution interpolant to construct trial and weight functions, a wavelet multiresolution interpolation Galerkin method (WMIGM) is proposed for solving elasticity problems. In this WMIGM, the essential boundary conditions can be imposed with ease as in the conventional finite element method. The stiffness matrix can be efficiently obtained through semi-analytical integration using an underlying general database, instead of the numerical integration usually requiring a mesh. The accuracy of the WMIGM is examined through theoretical analysis and benchmark problems. Results demonstrate that the proposed WMIGM has an excellent accuracy, optimal rate of convergence and competitive efficiency, as well as an excellent stability against irregular nodal distribution. Most importantly, by adding more nodes into local region only, a high resolution of localized steep gradients can be achieved as desired without changing the existing nodes.

Appendices

Appendix A: Proof of the decomposition (2.7)

Substituting the two-scale relation (2.3) into the approximation (2.6), yields

$$ S^{j} f({\mathbf{x}}) = \sum\limits_{k \in Even} {\sum\limits_{l \in Even} {f({\mathbf{x}}_{j + 1,k,l} )\theta_{j + 1,k,l} ({\mathbf{x}})} } + \sum\limits_{k \in Even} {\sum\limits_{l \in Odd} {[\sum\limits_{m \in Odd} {p_{m} f({\mathbf{x}}_{j + 1,k,l - m} )} ]\theta_{j + 1,k,l} ({\mathbf{x}})} } + \sum\limits_{k \in Odd} {\sum\limits_{l \in Even} {\left[ {\sum\limits_{n \in Odd} {p_{n} f({\mathbf{x}}_{j + 1,k - n,l} )} } \right]\theta_{j + 1,k,l} ({\mathbf{x}})} } + \sum\limits_{k \in Odd} {\sum\limits_{l \in Odd} {\left[ {\sum\limits_{n \in Odd} {\sum\limits_{m \in Odd} {p_{n} p_{m} f({\mathbf{x}}_{j + 1,k - n,l - m} )} } } \right]\theta_{j + 1,k,l} ({\mathbf{x}})} } $$

(A.1)

in which the property $p_{k} = \delta_{0,k}$ for $k \in Even$ is considered. Following this property and Eq. (2.3), there is the relation

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

(A.2)

where $k \in Even$ and $l \in Odd$. By using the same method, we also can obtain the relations

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

(A.3)

and

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

(A.4)

Substituting Eqs. (A.2–A.4) into Eq. (A.1), yields

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

(A.5)

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

Finally, we can obtain $R^{j} f({\mathbf{x}})$ defined by Eq. (2.8) by subtracting $S^{j} f({\mathbf{x}})$ given by Eq. (A.5) from $S^{j + 1} f({\mathbf{x}})$ shown in Eq. (2.6).

Appendix B: Proof of Theorem 2.2

Since $f({\mathbf{x}})$ is sufficiently smooth in the domain $\Omega$, one always can find a proper supplement of the definition of function $f({\mathbf{x}})$ in the domain $\bar{\Omega }$ such that $f({\mathbf{x}})$ is smooth enough in the domain $\Omega \cup \bar{\Omega }$ [54]. Then following Eqs. (2.12–2.15), we have

$$ \begin{aligned} S_{L}^{j} f({\mathbf{x}}) & = \sum\limits_{l = 1}^{{N_{j} }} {f({\mathbf{x}}_{l} )\theta_{{j,kx({\mathbf{x}}_{l} ),ky({\mathbf{x}}_{l} )}} ({\mathbf{x}})} + \sum\limits_{{l = - N_{e} }}^{ - 1} {[f({\mathbf{x}}_{l} ) + \tilde{f}_{l} ({\mathbf{x}}_{l} ) - f({\mathbf{x}}_{l} )]\theta_{{j,kx({\mathbf{x}}_{l} ),ky({\mathbf{x}}_{l} )}} ({\mathbf{x}})} \\ & = S^{j} f({\mathbf{x}}) + \sum\limits_{{l = - N_{e} }}^{ - 1} {[\tilde{f}_{l} ({\mathbf{x}}_{l} ) - f({\mathbf{x}}_{l} )]\theta_{{j,kx({\mathbf{x}}_{l} ),ky({\mathbf{x}}_{l} )}} ({\mathbf{x}})} . \\ \end{aligned} . $$

(B.1)

By applying Theorem 2.1 and Eq. (B.1), one can obtain

$$ ||S_{L}^{j} f({\mathbf{x}}) - f({\mathbf{x}})||_{\infty } = ||S^{j} f({\mathbf{x}}) - f({\mathbf{x}})||_{\infty } + \sum\limits_{{l = - N_{e} }}^{ - 1} {||[\tilde{f}_{l} ({\mathbf{x}}_{l} ) - f({\mathbf{x}}_{l} )]\theta_{{j,kx({\mathbf{x}}_{l} ),ky({\mathbf{x}}_{l} )}} ({\mathbf{x}})||_{\infty } } \le C_{1,\infty } 2^{ - j\gamma } + \sum\limits_{{l = - N_{e} }}^{ - 1} {||[\tilde{f}_{l} ({\mathbf{x}}_{l} ) - f({\mathbf{x}}_{l} )]\theta_{{j,kx({\mathbf{x}}_{l} ),ky({\mathbf{x}}_{l} )}} ({\mathbf{x}})||_{\infty } } ,\quad {\text{for}}\quad x \in \Omega . $$

(B.2)

Since $\tilde{f}_{l} ({\mathbf{x}})$ is a Lagrange interpolation with order $\eta_{l}$ of function $f({\mathbf{x}})$, by using the relations $|x_{l} - \tilde{x}_{l,i} | \le \zeta_{1} 2^{ - j}$ and $|y_{l} - \tilde{y}_{l,i} | \le \zeta_{2} 2^{ - j}$ given in Sect. 2.2, we have [56, 57]

$$ |f({\bar{\mathbf{x}}}_{l} ) - \tilde{f}_{l} ({\bar{\mathbf{x}}}_{l} )| \le \varepsilon_{2} 2^{{ - j\eta_{l} }} $$

(B.3)

in which $\varepsilon_{2}$ is a constant with the property $\varepsilon_{2} = 0$ for $D_{{\eta_{l} }}^{0} f({\mathbf{x}}) = D_{{\eta_{l} }}^{{\eta_{l} }} f({\mathbf{x}}) \equiv 0$. On the other hand, considering $\theta_{j,k,l} ({\mathbf{x}}) \in C_{0}^{\gamma /2 - 1} \Omega_{j,k,l}^{\inf }$, one can obtain

$$ \theta_{j,k,l} ({\mathbf{x}}) = 0,\quad {\text{for}}\quad k \notin (2^{j} x - \gamma + 1,2^{j} x + \gamma - 1)\quad {\text{or}}\quad l \notin (2^{j} y - \gamma + 1,2^{j} y + \gamma - 1) $$

(B.4)

which implies that there are at most $(2\gamma - 3)^{2}$ wavelet basis functions $\theta_{j,k,l} ({\mathbf{x}}) \ne 0$ for any point x. Therefore, Eq. (B.2) can be expressed as

$$ ||S_{L}^{j} f({\mathbf{x}}) - f({\mathbf{x}})||_{\infty } \le C_{1,\infty } 2^{ - j\gamma } + (2\gamma - 3)^{2} \varepsilon_{2} 2^{ - j\eta } \le C_{2,\infty } 2^{ - j\lambda } \quad {\text{for}}\quad x \in \Omega $$

(B.5)

where the constant $C_{2,\infty } = 0$, if $D_{\lambda }^{0} f({\mathbf{x}}) = D_{\lambda }^{\lambda } f({\mathbf{x}}) \equiv 0$. By applying Eq. (B.5), Eq. (2.18) can be obtained readily [54, 55].

Appendix C: Proof of Proposition 2.1

By applying the property of the modified wavelet basis $\varphi_{j,l} ({\mathbf{x}}_{k} ) = \delta_{k,l}$ for $k,l = 1,2, \ldots N_{i}$ given in Sect. 2.2, we can obtain directly

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

(C.1)

On the other hand, we assume that for $j_{0} \le o < J$ there is the relation

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

(C.2)

Then following Eqs. (2.20–2.22), we have

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

(C.3)

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

Due to $\theta (q) = \delta_{0,q}$ for all $q \in {\mathbb{Z}}$, there is $\theta_{o + 1,k,l} ({\mathbf{x}}_{j,n,m} ) = \theta (2^{o + 1 - j} n - k)\theta (2^{o + 1 - j} m - l) = 0$ for all nodes ${\mathbf{x}}_{j,n,m}$, $j_{0} \le j \le o$, because both of $2^{o + 1 - j} n$ and $2^{o + 1 - j} m$ are even numbers, and at least one of k and l is odd number. Therefore, Eq. (C.3) and the assumption (C.2) state

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

(C.4)

For $j = o + 1$, by using the property $\theta_{o + 1,k,l} ({\mathbf{x}}_{o + 1,n,m} ) = \delta_{n,k} \delta_{m,l}$, Eq. (C.3) can be reduced into

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

(C.5)

Equations (C.4, C.5) yield

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

(C.6)

Finally, one can obtain Proposition 2.1 by combining the recurrence relations (C.2, C.6) with the starting condition (C.1).

Appendix D: Proof of Theorem 2.3

Following the relations (2.19, 2.20), for $n > j_{0}$ there is the relation

$$ ||f({\mathbf{x}}) - P_{{j_{0} }}^{n} f({\mathbf{x}})||_{\infty } \le ||f({\mathbf{x}}) - P_{{j_{0} }}^{n - 1} f({\mathbf{x}})||_{\infty } + \sum\limits_{{(k,l) \in \Re_{n - 1} }} {|f({\mathbf{x}}_{j + 1,k,l} ) - P_{{j_{0} }}^{n - 1} f({\mathbf{x}}_{j + 1,k,l} )| \times ||\theta_{j + 1,k,l} ({\mathbf{x}})||_{\infty } } . . $$

(D.1)

By applying $\theta_{j,k,l} ({\mathbf{x}}) \in C_{0}^{\gamma /2 - 1} \Omega_{j,k,l}^{\inf }$ and Eq. (B.4), Eq. (D.1) can be expressed as

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

(D.2)

in which $\varepsilon_{3}$ is a constant. By using Eq. (D.2) iteratively, and considering $P_{{j_{0} }}^{{j_{0} }} f({\mathbf{x}}) = S_{L}^{{j_{0} }} f({\mathbf{x}})$ and Theorem 2.2, we have

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

(D.3)

where $\varepsilon_{4}$ is a constant, and the constant $C_{3,\infty } = 0$ if $D_{\lambda }^{0} f({\mathbf{x}}) = D_{\lambda }^{\lambda } f({\mathbf{x}}) \equiv 0$. On the basis of Eq. (D.3), it is very easy to check Eq. (2.23), as well as Eq. (2.24) [54, 55].

Appendix E: Evaluation of connection coefficients

Since $\theta^{(\alpha )} (x) = 0$, $\alpha = 0,1, \ldots ,\gamma /2$ for $|x| > \gamma - 1$, by directly examining the definition (4.23) of the three-term multiresolution connection coefficient $\Gamma_{k,l}^{n,m} (j_{1} ,j_{2} ,x)$, one can obtain directly

$$ \Gamma_{k,l}^{n,m} (j_{1} ,j_{2} ,x) = \Gamma_{k,l}^{n,m} (j_{1} ,j_{2} ,\gamma - 1),\quad {\text{for}}\quad x \ge \gamma - 1 $$

(E.1)

and

$$ \Gamma_{k,l}^{n,m} (j_{1} ,j_{2} ,x) = 0,\quad {\text{for}}\quad x \le 1 - \gamma ,\quad {\text{or}}\quad |k| \ge (2^{{j_{1} }} + 1)(\gamma - 1),\quad {\text{or}}\quad |l| \, \ge (2^{{j_{2} }} + 1)(\gamma - 1). $$

(E.2)

Then, substituting the two-scale relation (2.3) into Eq. (4.23), and applying Eq. (E.2), we have

$$ \begin{aligned} \Gamma_{k,l}^{n,m} (j_{1} ,0,x) & = 2^{(n + m - 1)} \sum\limits_{o = 1 - \gamma }^{\gamma - 1} {\sum\limits_{q = 1 - \gamma }^{\gamma - 1} {p_{o} p_{q} \int_{1 - \gamma }^{2x - o} {\frac{{d^{n} \theta (y)}}{{dy^{n} }}\frac{{d^{m} \theta [2^{{j_{1} - 1}} y - (k - 2^{{j_{1} - 1}} o)]}}{{dy^{m} }}\theta [y - (2l + q - o)]dy} } } \\ & = 2^{(n + m - 1)} \sum\limits_{o = 1 - \gamma }^{\gamma - 1} {\sum\limits_{q = 1 - \gamma }^{\gamma - 1} {p_{o} p_{q} \Gamma_{{k - 2^{{j_{1} - 1}} o,2l + q - o}}^{n,m} (j_{1} - 1,0,2x - o)} } \\ \end{aligned} $$

(E.3)

and

$$ \Gamma_{k,l}^{n,m} (j_{1} ,j_{2} ,x) = 2^{(n + m - 1)} \sum\limits_{o = 1 - \gamma }^{\gamma - 1} {\sum\limits_{q = 1 - \gamma }^{\gamma - 1} {p_{o} p_{q} \Gamma_{{2k + q - 2^{{j_{1} }} o,l - 2^{{j_{2} - 1}} o}}^{n,m} (j_{1} ,j_{2} - 1,2x - o)} } . $$

(E.4)

One can see that the coefficients $\Gamma_{k,l}^{n,m} (j_{1} ,0,x)$ for $j_{1} \ge 1$ can be obtained readily by applying Eq. (E.3) iteratively, when the coefficients $\Gamma_{k,l}^{n,m} (0,0,x)$ are known. Then, the connection coefficients $\Gamma_{k,l}^{n,m} (j_{1} ,j_{2} ,x)$ for $j_{1} ,j_{2} \ge 0$ and $k,l \in {\mathbb{Z}}$ can be evaluated through $\Gamma_{k,l}^{n,m} (j_{1} ,0,x)$ and Eq. (E.4).

Further, the coefficient $\Gamma_{k,l}^{n,m} (0,0,x)$ can be expressed as

$$ \Gamma_{k,l}^{n,m} (0,0,x) = \Gamma_{ - l,k - l}^{n,m} (x - l) - \Gamma_{ - l,k - l}^{n,m} (1 - \gamma - l) $$

(E.5)

where the classical three-term connection coefficient [52, 53]

$$ \Gamma_{k,l}^{n,m} (x) = \int_{1 - \gamma }^{x} {\theta (y)\theta^{(n)} (y - k)\theta^{(m)} (y - l)dy} $$

(E.6)

which possesses the properties [52, 53]

$$ \Gamma_{k,l}^{n,m} (x) = \Gamma_{k,l}^{n,m} (\gamma - 1),\quad {\text{for}}\quad x \ge \gamma - 1 $$

(E.7)

and

$$ \Gamma_{k,l}^{n,m} (x) = 0,\quad {\text{for}}\quad x \le 1 - \gamma \quad {\text{or}}\quad |k| \ge 2\gamma - 2\quad {\text{or}}\quad |l| \, \ge 2\gamma - 2\quad {\text{or}}\quad k \ge x + \gamma - 1\quad {\text{or}}\quad l \ge x + \gamma - 1. $$

(E.8)

Substituting the two-scale relation (2.3) into Eq. (E.6), and applying Eq. (E.8), yields

$$ \begin{aligned} \Gamma_{k,l}^{n,m} (x) & = 2^{n + m} \sum\limits_{i = 1 - \gamma }^{\gamma - 1} {\sum\limits_{j = 1 - \gamma }^{\gamma - 1} {\sum\limits_{o = 1 - \gamma }^{\gamma - 1} {p_{i} p_{j} p_{o} } } } \int_{1 - \gamma }^{x} {\theta (2y - i)\theta^{(n)} (2y - 2k - j)\theta^{(m)} (2y - 2l - o)dy} \\ & = 2^{n + m - 1} \sum\limits_{i = 1 - \gamma }^{\gamma - 1} {\sum\limits_{j = 3 - 2\gamma }^{2\gamma - 3} {\sum\limits_{o = 3 - 2\gamma }^{2\gamma - 3} {p_{i} p_{j + i - 2k} p_{o + i - 2k} \Gamma_{j,o}^{n,m} (2x - i)} } } . \\ \end{aligned} $$

(E.9)

It can be found that the values of $\Gamma_{k,l}^{n,m} (x)$ at all dyadic points $x = k/2^{j}$ can be obtained readily, if the values of $\Gamma_{k,l}^{n,m} (x)$ values at integer points are known. These values at integer points can be evaluated by using a modified version of the algorithm which is developed for orthogonal wavelets by Chen et al. [52] and Zhang et al. [53]. In the following, an alternative method which is much easier to implement compared with the existing algorithm [52, 53] will be proposed. This algorithm also can be directly extended to calculate the three-term connection coefficient for other wavelets.

For integer x, there is the relation

$$ \begin{aligned} \Gamma_{k,l}^{n,m} (x) & = \sum\limits_{i = 1 - \gamma }^{x - 1} {\int_{i}^{i + 1} {\theta (y)\theta^{(n)} (y - k)\theta^{(m)} (y - l)dy} } \\ & = \sum\limits_{i = 1 - \gamma }^{x - 1} {\int_{0}^{1} {\theta (y + i)\theta^{(n)} (y - k + i)\theta^{(m)} (y - l + i)dy} } = \sum\limits_{i = 1 - \gamma }^{x - 1} {\Lambda_{ - i,k - i,l - i}^{n,m} } \\ \end{aligned} $$

(E.10)

where the integral

$$ \Lambda_{i,j,k}^{n,m} = \int_{0}^{1} {\theta (y - i)\theta^{(n)} (y - j)\theta^{(m)} (y - k)dy} . $$

(E.11)

It can be seen from Eq. (E.10) that the values of $\Gamma_{k,l}^{n,m} (x)$ at integer points can be obtained readily, when the integrals $\Lambda_{i,j,k}^{n,m}$ are known. Considering $\theta^{(\alpha )} (x) = 0$, $\alpha = 0,1, \ldots ,\gamma /2$ for $|x| > \gamma - 1$, it is easy to check the relation

$$ \Lambda_{i,j,k}^{n,m} = 0,\quad {\text{for}}\quad i,j,{\text{or }}k \notin \{ 2 - \gamma ,3 - \gamma , \ldots ,\gamma - 1\} . $$

(E.12)

Substituting the two-scale relation (2.3) into Eq. (E.11), and applying Eq. (E.12), one can obtain

$$ \begin{aligned} \Lambda_{i,j,k}^{n,m} & = 2^{n + m} \sum\limits_{l = 1 - \gamma }^{\gamma - 1} {\sum\limits_{o = 1 - \gamma }^{\gamma - 1} {\sum\limits_{q = 1 - \gamma }^{\gamma - 1} {p_{l} p_{o} p_{q} \int_{0}^{1} {\theta (2y - 2i - l)\theta^{(n)} (2y - 2j - o)\theta^{(m)} (2y - 2k - q)dy} } } } \\ & = 2^{n + m - 1} \sum\limits_{l = 2 - \gamma }^{\gamma - 1} {\sum\limits_{o = 2 - \gamma }^{\gamma - 1} {\sum\limits_{q = 2 - \gamma }^{\gamma - 1} {p_{l - 2i} p_{o - 2j} p_{q - 2k} \int_{0}^{2} {\theta (y - l)\theta^{(n)} (y - o)\theta^{(m)} (y - q)dy} } } } \\ & = 2^{n + m - 1} \sum\limits_{l = 2 - \gamma }^{\gamma - 1} {\sum\limits_{o = 2 - \gamma }^{\gamma - 1} {\sum\limits_{q = 2 - \gamma }^{\gamma - 1} {(p_{l - 2i} p_{o - 2j} p_{q - 2k} + p_{l - 2i + 1} p_{o - 2j + 1} p_{q - 2k + 1} )\Lambda_{l,o,q}^{n,m} } } } . \\ \end{aligned} $$

(E.13)

Letting $i,j,{\text{and }}k = 2 - \gamma ,3 - \gamma , \ldots ,\gamma - 1$, respectively in Eq. (E.13), yields

$$ {\mathbf{B\Lambda }}^{n,m} = {\varvec{\Lambda}}^{n,m} $$

(E.14)

in which the matrix ${\mathbf{B}} = \{ b_{gh} = 2^{n + m - 1} (p_{l - 2i} p_{o - 2j} p_{q - 2k} + p_{l - 2i + 1} p_{o - 2j + 1} p_{q - 2k + 1} )\}$, the vector ${\varvec{\Lambda}}^{n,m} = \{ \Lambda_{g}^{n,m} = \Lambda_{i,j,k}^{n,m} \}^{\text{T}}$, and subscripts $g = (i + \gamma - 2)(2\gamma - 2)^{2} + (j + \gamma - 2)(2\gamma - 2) + k + \gamma - 1$ and $h = (l + \gamma - 2)(2\gamma - 2)^{2} + (o + \gamma - 2)(2\gamma - 2) + q + \gamma - 1$.

One can see from Eq. (E.14) that ${\varvec{\Lambda}}^{n,m}$ is an eigenvector of the matrix B corresponding to the unit eigenvalue. It can be found by numerical experiment that there are $1 \le \bar{\mu } \le (n + m + 1)(n + m + 2)/2$ linearly independent eigenvectors ${\bar{\mathbf{\Lambda }}}_{o}^{n,m} = \{ \bar{\Lambda }_{o,i,j,k}^{n,m} \}^{\text{T}}$, $o = 1,2, \ldots ,\bar{\mu }$, whose eigenvalues are all equivalent to 1. Thus, the integral $\Lambda_{i,j,k}^{n,m}$ can be expressed as

$$ \Lambda_{i,j,k}^{n,m} = \sum\limits_{o = 1}^{{\bar{\mu }}} {\lambda_{o} \bar{\Lambda }_{o,i,j,k}^{n,m} } $$

(E.15)

where the coefficients $\lambda_{o}$ for $o = 1,2, \ldots ,\bar{\mu }$ should be determined by additional constraint conditions.

Based on Theorem 2.1, one can obtain

$$ \sum\limits_{{i \in {\mathbb{Z}}}} {i^{a} \theta (x - i)} = x^{a} ,\quad \sum\limits_{{j \in {\mathbb{Z}}}} {j^{b} \theta^{(n)} (x - j)} = \frac{b!}{(b - n)!}x^{b - n} ,\quad \sum\limits_{{k \in {\mathbb{Z}}}} {k^{c} \theta^{(m)} (x - k)} = \frac{c!}{(c - m)!}x^{c - m} $$

(E.16)

where $a,b,c = 0,1, \ldots ,\gamma - 1$ and $d! = \infty$ for $d < 0$. Based on Eq. (E.16), we have

$$ \sum\limits_{{i \in {\mathbb{Z}}}} {\sum\limits_{{j \in {\mathbb{Z}}}} {\sum\limits_{{k \in {\mathbb{Z}}}} {i^{a} j^{b} k^{c} \theta (x - i)\theta^{(n)} (x - j)\theta^{(m)} (x - k)} } } = \frac{b!c!}{(b - n)!(c - m)!}x^{a + b + c - n - m} . $$

(E.17)

Integrating both sides of Eq. (E.17) over [0, 1], yields

$$ \sum\limits_{i = 2 - \gamma }^{\gamma - 1} {\sum\limits_{j = 2 - \gamma }^{\gamma - 1} {\sum\limits_{k = 2 - \gamma }^{\gamma - 1} {i^{a} j^{b} k^{c} \Lambda_{i,j,k}^{n,m} } } } = \frac{b!c!}{(a + b + c - n - m + 1)(b - n)!(c - m)!},\quad {\text{for}}\quad a,b,c = 0,1, \ldots ,\gamma - 1. $$

(E.18)

Substituting Eq. (E.15) into Eq. (E.18), gives

$$ \sum\limits_{o = 1}^{{\bar{\mu }}} {\lambda_{o} } \sum\limits_{i = 2 - \gamma }^{\gamma - 1} {\sum\limits_{j = 2 - \gamma }^{\gamma - 1} {\sum\limits_{k = 2 - \gamma }^{\gamma - 1} {i^{a} j^{b} k^{c} \bar{\Lambda }_{o,i,j,k}^{n,m} } } } = \frac{b!c!}{(a + b + c - n - m + 1)(b - n)!(c - m)!},\quad {\text{for}}\quad a,b,c = 0,1, \ldots ,\gamma - 1. $$

(E.19)

We note that Eq. (E.19) includes $\gamma^{3}$ equations for coefficients $\lambda_{o}$, $o = 1,2, \ldots ,\bar{\mu }$, which is an over-determined linear system, since $\bar{\mu } \le (n + m + 1)(n + m + 2)/2 \le (\gamma + 1)(\gamma + 2)/2 < \gamma^{3}$. But the numerical experiment shows that this over-determined linear system is compatible. Therefore, the coefficients $\lambda_{o}$ can be determined by using the least square method to solve Eq. (E.19). Once these coefficients are obtained, the integral $\Lambda_{i,j,k}^{n,m}$ can be evaluated through Eq. (E.15). Then, the values of $\Gamma_{k,l}^{n,m} (x)$ at integer points can be calculated by Eq. (E.10). Based on these basic data values, one can obtain the three-term multiresolution connection coefficient $\Gamma_{k,l}^{n,m} (j_{1} ,j_{2} ,x)$ with ease, by using the relations (E.9, E.5, E.3, E.4) in sequence.

For the two-term multiresolution connection coefficient ${\rm T}_{j,k}^{n,m} (x)$ defined by Eq. (4.24), by considering $\theta^{(\alpha )} (x) = 0$, $\alpha = 0,1, \ldots ,\gamma /2$ for $|x| > \gamma - 1$, there are also the following relations

$$ {\rm T}_{j,k}^{n,m} (x) = {\rm T}_{j,k}^{n,m} (\gamma - 1),\quad {\text{for}}\quad x \ge \gamma - 1 $$

(E.20)

and

$$ {\rm T}_{j,k}^{n,m} (x) = 0,\quad {\text{for}}\quad x \le 1 - \gamma ,\quad {\text{or}}\quad |k| \ge (2^{j} + 1)(\gamma - 1). $$

(E.21)

By considering $\theta (x) = 0$ for $|x| > \gamma - 1$, Eq. (E.16) gives

$$ 1 = \sum\limits_{l = 3 - 2\gamma }^{[x] + \gamma } {\theta (y - l)} ,\quad {\text{for}}\quad y \in [1 - \gamma ,x] $$

(E.22)

where $[x]$ is the nearest integer to x. Substituting Eq. (E.22) into Eq. (4.24), yields

$$ {\rm T}_{j,k}^{n,m} (x) = \sum\limits_{l = 3 - 2\gamma }^{[x] + \gamma } {\int_{1 - \gamma }^{x} {\theta^{(n)} (y)\theta^{(m)} (2^{j} y - k)\theta (y - l)dy} } = \sum\limits_{l = 3 - 2\gamma }^{[x] + \gamma } {\Gamma_{k,l}^{n,m} (j,0,x)} . $$

(E.23)

It can be seen that the two-term multiresolution connection coefficient ${\rm T}_{j,k}^{n,m} (x)$ can be obtained based on Eq. (E.23) and the three-term multiresolution connection coefficient $\Gamma_{k,l}^{n,m} (j_{1} ,j_{2} ,x)$.