In this study, we examine biorthogonal wavelets that are tailored to a specific discrete pseudo-differential equation of the form $T_{\sigma}u = f$, where $T_{\sigma}$ is an invertible discrete pseudo-differential operator defined on the lattice $\mathbb{Z}^{n}$ for every $f\in\ell^{2}(\mathbb{Z}^{n})$. Our focus is on computing Galerkin approximations of the solution to this problem using an adaptive algorithm.
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