Analytic wavelets generated by radial functions
The paper deals with a class of non-stationary multiresolution analysis and wavelets generated by radial basis functions. The multiresolution analysis is generated by \(\phi \in L^2(\mathbb{R}^d)\) by letting \[ V_j := \text{clos}_{L^2} \text{span} \{\phi (\cdot - A^{-j}k), \quad k \in\mathbb{Z}\}, \] where \(A\) is a \(d \times d\) integer matrix, and all eigenvalues of \(A\) have absolute value \(> 1\). Note that no dilation of \(\phi\) appears in the definition of \(V_j\). The Littlewood-Paley identity is proved, and time and frequency localization of the corresponding wavelet basis are studied. Examples are based on multiquadric wavelets.
scientific article; zbMATH DE number 892179
scientific article