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Tensorproduct Monotonicity Preservation
"... Abstract: This paper studies systems of tensorproduct functions for which the functions they span are monotonic in any direction when their control nets are monotonic in that direction. It is shown that Bernstein polynomials and Bsplines have this property but that totally positive systems in gene ..."
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Abstract: This paper studies systems of tensorproduct functions for which the functions they span are monotonic in any direction when their control nets are monotonic in that direction. It is shown that Bernstein polynomials and Bsplines have this property but that totally positive systems
Hierarchical Kronecker tensorproduct approximations
 MATHEMATIK IN DEN NATURWISSENSCHAFTEN, LEIPZIG, PREPRINT NO
, 2003
"... The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integrodifferential) operators, especially for highdimensional problems. These new formats elaborate on ..."
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Cited by 42 (23 self)
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” or hierarchical matrices (cf. [17, 18]). We give a proof for the existence of such formats and expound a gainful combination of the Kroneckertensorproduct structure and the arithmetic for hierarchical matrices.
Optimized TensorProduct Approximation Spaces
"... . This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the approximation of the embedding of the intersection of classes of functions with bounded mixed derivatives in standard S ..."
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Cited by 42 (15 self)
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Sobolev spaces. Based on the framework of tensorproduct biorthogonal wavelet bases and stable subspace splittings, the problem is reduced to diagonal mappings between Hilbert sequence spaces. We construct operator adapted finiteelement subspaces with a lower dimension than the standard fullgrid spaces
Tensorproduct adaptive grids based on coordinate transformations
 J. OF COMP. APPL. MATHS
, 2004
"... ..."
Shape Properties of TensorProduct Bernstein Polynomials
"... . This paper reviews recently established shape properties of tensorproduct Bernstein polynomials, emphasizing monotonicity, convexity, and sign changes. Some comparisons with other tensorproduct bases are also discussed. 1. Introduction This chapter concerns shape properties of the tensorproduct ..."
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surface. If one replaced the coefficients c ij by control points c ij in R 3 then f would become a B'ezier surface. B'ezier surfaces (and their generalizations to spline surfaces) are better suited to surface design. One has much greater freedom to model complicated shapes in the parametric
Bases and dimensions of bivariate hierarchical tensor–product splines
"... We prove that the dimension of bivariate tensor–product spline spaces of bi– degree (d,d) with maximum order of smoothness on a multi–cell domain (more precisely, on a set of cells from a tensor–product grid) is equal to the number of tensor–product B–spline basis functions, defined by only single k ..."
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We prove that the dimension of bivariate tensor–product spline spaces of bi– degree (d,d) with maximum order of smoothness on a multi–cell domain (more precisely, on a set of cells from a tensor–product grid) is equal to the number of tensor–product B–spline basis functions, defined by only single
Approximation of the Electron Density of Aluminium Clusters in TensorProduct Format
, 2009
"... cMaxPlanckInstitute for Mathematics in the Sciences, ..."
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cMaxPlanckInstitute for Mathematics in the Sciences,
Approximation of the Electron Density of Aluminium Clusters in TensorProduct Format
"... cMaxPlanckInstitute for Mathematics in the Sciences, ..."
On the completeness of hierarchical tensorproduct Bsplines
"... Given a grid in Rd, consisting of d biinfinite sequences of hyperplanes (possibly with multiplicities) orthogonal to the d axes of the coordinate system, we consider the spaces of tensorproduct spline functions of a given degree on a multicell domain. Such a domain consists of finite set of cell ..."
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of cells which are defined by the grid. A piecewise polynomial function belongs to the spline space if its polynomial pieces on adjacent cells have a contact according to the multiplicity of the hyperplanes in the grid. We prove that the connected components of the associated set of tensorproduct B
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