Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47887220210701Multi-dimensional wavelets on Sobolev spaces21121813007510.22034/kjm.2021.202782.1576ENFatemehEsmaeelzadehIslamic Azad University, Bojnourd BranchJournal Article20190924In this paper, for admissible and integrable function $psi$ in $L^2(mathbb{R}^n)$, the multi-dimensional continuous wavelet transform on Sobolev spaces is defined. The inversion formula for this transform on Sobolev spaces is established and as a result it is concluded that there is an isometry of Sobolev spaces $H_s(mathbb{R}^n)$ into $H_{0,s}(mathbb{R}^n times mathbb{R}^+_0times S^{n-1})$, for arbitrary real $s$. Also, among other things, it is shown that the range of this transform is a reproducing kernel Hilbert space and the reproducing kernel is found.http://www.kjm-math.org/article_130075_9a177af8e6f9787ecd973580817508db.pdf