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On the Evaluation of Box Splines
, 1993
"... the rule M \Xi : C(IR s ) ! IR : ' 7! hM \Xi ; 'i := Z '(\Xit)dt: Here, := [0 : : 1) n is the halfopen unit cube or `box' in IR n . The distribution M \Xi is always representable as an L1 function on ran \Xi. In particular, if \Xi is of full rank, i.e., ran \Xi = IR s , then M \Xi i ..."
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Cited by 158 (10 self)
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the rule M \Xi : C(IR s ) ! IR : ' 7! hM \Xi ; 'i := Z '(\Xit)dt: Here, := [0 : : 1) n is the halfopen unit cube or `box' in IR n . The distribution M \Xi is always representable as an L1 function on ran \Xi. In particular, if \Xi is of full rank, i.e., ran \Xi = IR s , then M \Xi is (represented by) an L1 (IR s )function, and this function is piecewise polynomial (=: pp), of exact degree n \Gamma s, with support the compact, convex set \Xi . The breakplanes of this pp function are given by certain translates of the hyperplanes spanned
Fast Fourier transforms for nonequispaced data: A tutorial
, 2000
"... In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity o ..."
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Cited by 111 (33 self)
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In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTalgorithms with respect to roundoff errors and apply NDFTalgorithms for the fast computation of Bessel transforms.
1 On the evaluation of box splines
"... The rst (and for some still the only) multivariate Bspline is what today one would call the simplex spline, since it is derived from a simplex, and in distinction to other polyhedral splines, such as the cone spline and the box spline. The simplex spline was rst talked about in 1976. However, it wa ..."
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The rst (and for some still the only) multivariate Bspline is what today one would call the simplex spline, since it is derived from a simplex, and in distinction to other polyhedral splines, such as the cone spline and the box spline. The simplex spline was rst talked about in 1976. However, it was only after Micchelli [Micchelli, 1980] established recurrence relations for them that the topic of simplex splines and other multivariate Bsplines really took o. Their cousins, the box splines, were thought particularly attractive because their recurrence relations turned out to be very simple indeed. It was, therefore, a shock to me when, in the process of doing my bit on the book [de Boor, Hollig, Riemenschneider, 1993], I found that it was nontrivial to make e ective use of these recurrence relations. It is one purpose of this note to relate my di culties and how I tried to deal with them. This is not the place to giveanintroduction to box spline theory, nor to review the relevant literature. Rather, the reader is urged to consult [de Boor, Hollig, Riemenschneider, 1993] for missing details and the proper literature references (as well as for many illustrations, two of which are reproduced below). 1 Box splines de ned The svariate box spline M: = M ( j) is de ned in terms of its direction matrix 2 IR s n, (:;j) 2 IR s n0, j =1;:::;n, as the distribution or linear functional given by the rule M: C(IR s Z) ! IR: ' 7! hM;'i: = ' ( t)dt: