Multi-dimensional wavelets on Sobolev spaces

Document Type : Original Article


Islamic Azad University, Bojnourd Branch



‎In 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 $H_{0,s}(\mathbb{R}^n \times \mathbb{R}^+_0\times S^{n-1})$‎, ‎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.