Multivariate Periodic Wavelets of de la Vallée Poussin Type

What is it about?

In this paper we present a general approach to multivariate periodic wavelets generated by scaling functions of de la Vallée Poussin type. These scaling func- tions and their corresponding wavelets are determined by their Fourier coeffi- cients, which are samples values of a function, that can be chosen arbitrarily smooth, even with different smoothness in each direction. This construction generalizes the one-dimensional de la Vallée Poussin means to the multivariate case and enables the construction of wavelet systems, where the set of dilation matrices for the two-scale relation of two spaces of the multiresolution analysis may contain shear and rotation matrices. It further enables the functions con- tained in each of the function spaces from the corresponding series of scaling spaces to have a certain direction or set of directions as their focus, which is illustrated by detecting jumps of certain directional derivatives of higher order.

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