Wavelets Associated with Nonuniform Multiresolution Analysis on Positive Half Line

Abstract

Gabardo and Nashed considered a generalization of the notion of multiresolution analysis, which is called nonuniform multiresolution analysis (NUMRA) and is based on the theory of spectral pairs. In this set up, the associated subspace $V_{0}$ of $L^{2}(\mathbb {R})$ has, as an orthonormal basis, a collection of translates of the scaling function $\phi$ of the form $\{\phi (x-\lambda )\}_{\lambda \in \varLambda }$ where $\varLambda = \{0,r/N\}+ 2\mathbb {Z}$, $N\ge 1$ is an integer and r is an odd integer with $1\le r\le 2N-1$ such that r and N are relatively prime and $\mathbb {Z}$ is the set of all integers. The main results of Gabardo and Nashed deal with necessary and sufficient condition for the existence of associated wavelets and extension of Cohen’s theorem.