Shift-Invariant Spaces from Rotation-Covariant Functions

We consider shift-invariant multiresolution spaces generated by rotation-covariant functions ρ in $\mathbb{R}^{ 2 }$ . To construct corresponding scaling and wavelet functions, ρ has to be localized with an appropriate multiplier, such that the localized version is an element of $L ^{ 2 } (\mathbb{R} ^{ 2 } )$. We consider several classes of multipliers and show a new method to improve regularity and decay properties of the corresponding scaling functions and wavelets. The wavelets are complex-valued functions, which are approximately rotation-covariant and therefore behave as Wirtinger differential operators. Moreover, our class of multipliers gives a novel approach for the construction of polyharmonic B-splines with better polynomial reconstruction properties.