The wavelet-Galerkin laplacian, $\Delta$.

The wavelet-Galerkin Laplacian appears in the basic wave equation

$$\psi_{tt}=c^2\Delta \psi$$

and the spectra of $\Delta$, determines the dispersion relation. The wavelet-Galerkin laplacian is described by the following matrix equation

$$\Omega\Psi + \Psi\Omega$$

where $\Omega$ is a circulant matrix whose rows are periodically shifted copies of the two term connection coefficient vector $\Omega^{02}$

$$\Omega = circ\{\Omega_0,\Omega_1,\cdots,\Omega_p,0,\cdots,0,\Omega_{-p},\cdots, \Omega_{-1}\}$$

where

$$\Omega_j=\int\phi (x)\phi _{xx}(x-j)dx.$$

Note that $\Omega=\Omega^t$.

All circulant matrices commute and have a common set of eigenvectors. In effect,$\Omega$ can be diagonalized by

$$\Omega = \Phi D_{\Omega} \Phi^t$$

where $N=2m$

$$D_{\Omega}=diag\{\lambda_j\}$$

$$\lambda_j=\Omega_0 + 2\sum_{k=1}^p\Omega_k\cos(\frac{2\pi jk}{N})$$

and

$$\Phi=\frac{\sqrt{2}}{\sqrt{N}}\left( \begin{array}{llllllll} ... ...sqrt{2} &s_{(m-1)(N-1)} & \cdots & s_{(N-1)} \end{array}\right)$$

where $c_j=\cos{\frac{2\pi j}{N}}$ and $s_j=\sin{\frac{2\pi j}{N}}$. We note that $\Phi\Phi^t=I$. This procedure has been implemented using the Fast Fourier transform.

The following figures show the wavelet-Galerkin template and spectra for $-\frac{\partial}{\partial_x^2}$. We note that the essential support of the different templates are nearly identical.

The spectra for wavelet-Galerkin templates are equal to or greater than the exact Fourier spectra. In the opposite sense, the spectra for the finite differences defined by the cancelation of Taylor series error terms [4,17,10,22] are equal to or less than the exact Fourier spectra.

Figure 4: D6, D10, and D42 templates for $-\frac{\partial}{\partial_x^2}$

Figure 5: D6 to D32 wavelet-Galerkin, and Fourier, spectra for $-\frac{\partial}{\partial_x^2}$

Figure 6: D10, D20, D30 wavelet-Galerkin (green), Fourier (red), and Order 2, 4 , 6, 8, 10 finite difference (blue) spectra for $-\frac{\partial}{\partial_x^2}$