Convergence of Subdivision Schemes and Smoothness of Refinable Functions on p-adic Fields
Athira N, Lineesh M C
TL;DR
This work develops a theory of $p$-adic refinement equations and subdivision schemes with finitely supported masks on $I_p$, establishing $L_q$-convergence criteria through the $q$-norm joint spectral radius $\rho_q(\mathcal{A})$ of associated operators. By defining the subdivision operator $S_h$ and the family of matrices $A_{\epsilon}$, it connects the iteration $Q_h^n$ to spectral radii on finite-dimensional invariant subspaces, yielding necessary and sufficient convergence conditions in terms of $\rho_q(\{A_{\epsilon}|_V\})<p^{1/q}$. The paper also analyzes function smoothness on $\mathbb{Q}_p$, proving that $Lip(\alpha,L_q(\mathbb{Q}_p))=C^{\alpha}(\mathbb{Q}_p)$ for $\alpha>0$ and that every compactly supported $L_q$ function on $\mathbb{Q}_p$ is infinitely smooth, via the pseudo-differential operator $D^{\alpha}$. Together, these results extend classical Euclidean subdivision/smoothness theory to the $p$-adic setting and provide tools for constructing $p$-adic wavelets and MRAs with controlled convergence and regularity.
Abstract
A systematic and comprehensive study of p-adic refinement equations and subdivision scheme associated with a finitely supported refinement mask are carried out in this paper. The Lq -convergence of the subdivision scheme is characterized in terms of the q-norm joint spectral radii of a collection of operators associated with the refinement mask. Also, the smoothness of complex-valued functions on Qp is investigated.