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.


Published in:
Applied and Computational Harmonic Analysis, 25, 2, 240–265
Year:
2008
Publisher:
Elsevier
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 Record created 2008-12-10, last modified 2018-12-03

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