Multi-scale analysis and wavelet expansions of spaces of periodic distributions (Q1595669)

Let \({\mathcal D}\) denote the space of \(2\pi\)-periodic distributions and let \({\mathcal H}\) be the class of quasi-Banach spaces \(X\) which satisfy \(C^\infty\subset X\subset{\mathcal D}\) and other conditions concerning invariance under certain change of variables and pointwise multiplication by the functions \(\exp(inx)\), \(n\in\mathbb{Z}\). The author shows that if \(\{V^j\}^\infty_{j=0}\) is an arbitrary multi-scale analysis of the space \(X\in{\mathcal H}\), then there exists a family of functions \(\{\varphi^j\}^\infty_{j=0}\) such that the components of the vectors \(\varphi^\ell\) form bases of the spaces \(V^\ell\). Moreover, for any \(j_1\), \(j_2\), \(0\leq j_1< j_2\), we have: \[ \varphi^{j_1}(J_2x/J_2)= \sum^{J_2/J_1- 1}_{k=0} \varphi^{j_2}\Biggl(x+ {2\pi kJ_1\over J_2}\Biggr), \] where we use the convection that \(J_i= 2^{j_i}\) for \(i= 1,2\).