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The notion of vanishing-moment recovery (VMR) functions is introduced in this paper for the construction of compactly supported tight frames with two generators having the maximum order of vanishing moments as determined by the given refinable function, such as the mth order cardinal B-spline N m. Tight frames are also extended to " sibling frames " to(More)
We study Sobolev type estimates for the approximation order resulting from using strictly positive deenite kernels to do interpolation on the n-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger 10] and related estimates of 6]. These error estimates are then based on series expansions of(More)
Two simple constructive methods are presented to compute compactly supported tight wavelet frames for any given refinable function whose mask satisfies the QMF or sub-QMF conditions in the multivariate setting. We use one of our constructive methods in order to find tight wavelet frames associated with multivariate box splines, e.g., bivariate box splines(More)
where g ∈ L 2 (R) and M lβ T kα g(x) = e 2πilβx g(x − kα). In collaboration with K. Gröchenig, we proved that G(g, α, β) is a frame if αβ < 1 and g is a totally positive function of finite type, i.e., its Fourier transform is g(ω) = C m j=1 (1 + 2πiδ j ω) −1 , δ j ∈ R \ {0}. With T. Kloos from Dortmund, we found new properties of these frames: 1. The Zak(More)
Continuing our recent work in [5] we study polynomial masks of multivariate tight wavelet frames from two additional and complementary points of view: convexity and system theory. We consider such polynomial masks that are derived by means of the unitary extension principle from a single polynomial. We show that the set of such poly-nomials is convex and(More)
When a Cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L 2 = L 2 (IR) with dilation integer factor M ≥ 2, the standard " matrix extension " approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate(More)