Multi-dimensional Wavelets on Sobolev Spaces
Fatemeh Esmaeelzadeh · Khayyam journal of mathematics · 2021
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 $H_{0,s}(mathbb{R}^n times mathbb{R}^+_0times S^{n-1})$, 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.