Three term connection coefficients

Define the three term connection coefficients

$$\Omega_{jl}^{101} = \int\phi _{x}(x)\phi (x-j)\phi _{x}(x-l)dx$$

$$\Omega_{jl}^{000} = \int\phi (x)\phi (x-j)\phi (x-l)dx$$

With the summation convention on the indices $(j,l,n,m)$ the Wavelet-Galerkin equations with $C^2=1$ are

$$\Psi_{tt}(p,q)= \left(\Omega_{j,l}^{101}\Omega_{n,m}^{000}+ \Omega_{n,m}^{101}\Omega_{j,l}^{000}\right) K(j+p,n+q)\Psi(l+p,m+q)$$

Since $\sum_j\phi (x-j)=1$

$$\sum_j\Omega_{jl}^{000}=\int\phi (x)\phi (x-l)dx=\delta(l)$$

$$\sum_j\Omega_{jl}^{101}=\int\phi _x(x)\phi _x(x-l)dx=\Omega_l^{11}$$

For constant $k$ the wavelet-Galerkin equations reduce to convolutional form:

$$\Psi_{tt}(p,q)= \left(\Omega_{l}^{11}\delta(m)+ \Omega_{m}^{11}\delta(l)\right) k\Psi(l+p,m+q)$$

$$\Psi_{tt}(p,q)=k\left(\Omega_l^{11}\Psi(l+p,q)+\Omega_m^{11}\Psi(p,m+q)\right)$$

For piecewise constant $k$ the wavelet-Galerkin equations reduce to a convolutional form except for a region, $R$, of width $2m-1$ centered at a jump point, where $\Omega^{101}$ and $\Omega^{000}$ are of size $(m,m)$.

For $(p,q)$ not in $R$,

$$\Psi_{tt}(p,q)= k(p,q)\left(\Omega_l^{11}\Psi(l+p,q)+\Omega_m^{11}\Psi(p,m+q)\right)$$

For $(p,q)$ in $R$,

$$\Psi_{tt}(p,q)= \left(\Omega_{j,l}^{101}\Omega_{n,m}^{000}+ \Omega_{n,m}^{101}\Omega_{j,l}^{000}\right) k(j+p,n+q)\Psi(l+p,m+q)$$

If in $R$, the piecewise constant $k$ is locally a tensor product

$$k(p,q)=a(p)b(q)$$

where $a$ and $b$ are piecewise constant, then for $(p,q)$ in $R$,

$$\Psi_{tt}(p,q)= \left(\Omega_{j,l}^{101}\Omega_{n,m}^{000}+ \Omega_{n,m}^{101}\Omega_{j,l}^{000}\right) K(j+p,n+q)\Psi(l+p,m+q)$$

$$=(a(j+p)\Omega_{j,l}^{101}b(n+q)\Omega_{n,m}^{000} +a(j+p)\Omega_{j,l}^{000}b(n+q)\Omega_{n,m}^{101})\Psi(l+p,m+q)$$

$$=\left(A(p,l)B(q,m)+B(p,l)A(q,m)\right)\Psi(l+p,m+q)$$

Therefore in $R$, if piecewise constant $k$ is locally a tensor product

$$\Psi_{tt}(p,q)= \left(A(p,l)B(q,m)+B(p,l)A(q,m)\right)\Psi(l+p,m+q)$$