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On the construction of multivariate (pre)wavelets. (English) Zbl 0773.41013
Summary: A new approach for the construction of wavelets and prewavelets on \(\mathbb{R}^ d\) from multiresolution is presented. The method uses only properties of shift-invariant spaces and orthogonal projectors from \(L_ 2(\mathbb{R}^ d)\) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the natura of the intersection and the union of a scale of spaces to be used in a multiresolution.

MSC:
41A15 Spline approximation
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
41A30 Approximation by other special function classes
46C99 Inner product spaces and their generalizations, Hilbert spaces
42B99 Harmonic analysis in several variables
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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