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Shift-invariant Spaces of Multivariate Periodic Functions

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ASCW01 - Challenges in optimal recovery and hyperbolic cross approximation

One of the underlying ideas of multiresolution and wavelet analysis consists in the investigation of shift-invariant function spaces. In this talk one-dimensional shift-invariant spaces of periodic functions are generalized to multivariate shift-invariant spaces on non-tensor product patterns. These patterns are generated from a regular integer matrix. The decomposition of these spaces into shift-invariant subspaces can be discussed by the properties of these matrices. For these spaces we study different bases and their time-frequency localization. Of particular interest are multivariate orthogonal Dirichlet and de la Valle\'e Poussin kernels and the respective wavelets. This approach also leads to an adaptive multiresolution. Finally, with these methods we construct shearlets and show how we can detect jump discontinuities of given cartoon-like functions.

This talk is part of the Isaac Newton Institute Seminar Series series.

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