Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames (Q731376)

A sequence \(\{\Phi_{j-1}\}_{j\in\mathbb{N}}\), where \(\Phi_{j-1}\in L_2(\mathbb{R})\), is called sequence of nonstationary refinable functions, if \(\widehat\Phi_{j-1}(\xi)=\widehat a_j(\xi/2)\widehat\Phi_j(\xi/2)\) a.e. in \(\mathbb{R}\), \(j\in\mathbb{N}\), where \(\widehat a_j\) are \(2\pi\)-periodic measurable functions and \(~\widehat{}~\) means the Fourier transform. For \(\widehat b_j\) \(2\pi\)-periodic and measurable, the wavelet function \(\psi^l_{j-1}\), \(j\in\mathbb{N}\), \(l= 1,\dots, J_j\), are defined by formula \(\widehat\psi^l_{j-1}(\xi)= b^l_j(\xi/2)\phi_j(\xi/2)\). The main tool in the paper is obtained applying the notion of a nonstationary tight wavelet frame for \(L_2(\mathbb{R})\), \(X= X(\Phi_0,\{\psi_j^l\}_{j\in \mathbb{N}_0,l\in\{1,\dots,J_{j+1}\}})\). Let \(H^s(\mathbb{R})\) be the Sobolev space. Under some conditions on \(s\in\mathbb{R}\), there is defined an expression, equivalent to the norm of \(f\) in \(H^s(\mathbb{R})\) for all \(f\in H^s(\mathbb{R})\). This leads to the result that any compactly supported nonstationary wavelet frame can be property normalized into a pair of dual frames in the corresponding pair of Sobolev spaces.