2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106407We extend the existing theory of approximation orders provided by shift-invariant subspaces of $L_2$ to the setting of Sobolev spaces, provide treatment of $L_2$ cases that have not been covered before, and apply our results to determine approximation order of solutions to a refinement equation with a higher-dimensional solution space.49 pagesClassical Analysis and ODEsFunctional Analysis41A25, 41A63, 41A30, 41A15, 42B99, 46E35, 46E30Approximation orders of shift-invariant subspaces of $W^s_2({\Bbb R}^d)$text