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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)

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)

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)

In this paper a large class of universal windows for Gabor frames (Weyl-Heisenberg frames) is constructed. These windows have the fundamental property that every overcritical rectangular lattice generates a Gabor frame. Likewise, every undercritical rectangular lattice generates a Riesz sequence.

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)

1 Abstract. From the definition of tight frames, normalized with frame bound constant equal to one, a tight frame of wavelets can be considered as a natural generalization of an or-thonormal wavelet basis, with the only exception that the wavelets are not required to have norm equal to one. However, without the orthogonality property, the tight-frame… (More)

The notion of tight (wavelet) frames could be viewed as a generalization of orthonormal wavelets. By allowing redundancy, we gain the necessary flexibility to achieve such properties as " symmetry " for compactly supported wavelets and, more importantly, to be able to extend the classical theory of spline functions with arbitrary knots to a new theory of… (More)