Convergence of exponential sums that interpolate to Laplace transforms (1.1) have been studied by... more Convergence of exponential sums that interpolate to Laplace transforms (1.1) have been studied by several authors [3, 6, 8, 15]. For rational functions that interpolate to Markov functions (also called Hamburger or Stieltjes Series or Hilbert Transforms) (1.2) far more detailed convergence results are available (see [10, 11, 16] and references therein). Both (1.1) and (1.2) are special cases of the transform (1.3) where K(x, t) is a strictly totally positive kernel.
International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, 1990
Results from the analysis of binary subdivision schemes are reviewed with a focus on properties w... more Results from the analysis of binary subdivision schemes are reviewed with a focus on properties which are special to interpolatory schemes. Some new results are presented, in particular the known sufficient conditions for the convergence of binary subdivision schemes to Cv limit functions are proved to be necessary in the case of interpolatory schemes. Specific examples of interpolatory schemes are reviewed, and their properties are concluded from the general theory.
Dyn, N. and S. Rippa, Data-dependent triangulations for scattered data interpolation and finite e... more Dyn, N. and S. Rippa, Data-dependent triangulations for scattered data interpolation and finite element approximation, Applied Numerical Mathematics 12 (1993) 89-105. In this paper we review methods for improving the accuracy of approximation by piecewise linear polynomials defined over triangulations to bivariate functions, by adapting the triangulation to the structure of the approximated function. Many criteria for optimizing triangulations as well as efficient algorithms for computing data-dependent triangulations that are locally optimal with respect to the criteria are presented. It is demonstrated that data-dependent triangulations can improve significantly the quality of approximation and that long and thin triangles, which are traditionally avoided, are sometimes necessary to have. Our results are concerned with two processes of approximation: (i) scattered data interpolation to bivariate functions, (ii) finite element approximations to solutions of second-order elliptic partial differential equations. Extensions of these ideas to higher-order approximation schemes as well as extensions to multivariate approximation problems are also discussed.
We study properties of a binary operation between two compact sets depending on a weight in [0,1]... more We study properties of a binary operation between two compact sets depending on a weight in [0,1], termed metric average. The metric average is used in spline subdivision schemes for compact sets in |R^n, instead of the Minkowski convex combination of sets, to retain non-convexity, see N. Dyn, E. Farkhi, ``Spline subdivision schemes for compact sets with metric averages", Trends in Approximation Theory (2001). Some properties of the metric average of sets in |R, like the cancellation property and the linear behavior of the Lebesgue measure of the metric average with respect to the weight, are proven. We present an algorithm for computing the metric average of two compact sets in |R, which are finite unions of intervals, as well as an algorithm for reconstructing one of the metric average's operands, given the second operand, the metric average and the weight.
Curve subdivision is a technique for generating smooth curves from initial control polygons by re... more Curve subdivision is a technique for generating smooth curves from initial control polygons by repeated refinement. The most common subdivision schemes are based on linear refinement rules, which are applied separately to each coordinate of the control points, and the analysis of these schemes is well understood. Since the resulting limit curves are not sufficiently sensitive to the geometry of the control polygons, there is a need for geometric subdivision schemes. Such schemes take the geometry of the control polygons into account by using non-linear refinement rules and are known to generate limit curves with less artefacts. Yet, only few tools exist for their analysis, because the non-linear setting is more complicated. In this paper, we derive sufficient conditions for a convergent interpolatory planar subdivision scheme to produce tangent continuous limit curves. These conditions as well as the proofs are purely geometric and do not rely on any parameterization.
Interpolatory subdivision schemes generating surfaces from initial control nets with the topology... more Interpolatory subdivision schemes generating surfaces from initial control nets with the topology of a regular grid are introduced and analyzed. The schemes are based upon the butterfly-scheme and are shown to have components with continuous first order derivatives. The generalization of these schemes for the application to general control nets is also discussed. 1 Introduction The design of surfaces in CAGD starts with a set of points with connectivity relations between them, termed control points. The simplest connectivity is that of a regular grid, namely each point has two indices: p ij , i = 1; : : : ; N 1 , j = 1; : : : ; N 2 . Thus the four points p ij ; p i+1;j ; p i;j+1 and p i+1;j+1 constitute a "face", and each point p ij has four neighbors p i;j Sigma1 , p iSigma1;j . Another possible topology of the connectivity relations is that of a triangular grid, where each face is determined by three control points and each pair of control points can belong to at most tw...
Interpolatory subdivision schemes generating surfaces from initial control nets with the topology... more Interpolatory subdivision schemes generating surfaces from initial control nets with the topology of a regular grid are introduced and analyzed. The schemes are based upon the butterfly-scheme and are shown to have components with continuous first order derivatives. The generalization of these schemes for the application to general control nets is also discussed.
Given a set of matrices, modeled as samples of a matrix-valued function, we suggest a method to a... more Given a set of matrices, modeled as samples of a matrix-valued function, we suggest a method to approximate the underline function using a product approximation operator. This operator extends known approximation methods by exploiting the structure of the matrices in the samples set, and based on decomposition theorems. We introduce our approach in detail and discuss its advantages using a few examples. In addition, we provide basic tools for analyzing properties of the matrix functions generated by our approximation operators.
Dahmen and Micchelli [8] have shown that in general the coefficients of the refined control nets ... more Dahmen and Micchelli [8] have shown that in general the coefficients of the refined control nets of a box spline surface converge to the surface at (at least) the rate of the refinement. The purpose of this article is to show that under mild additional assumptions the convergence rate is even quadratic. Although this rate is in general best possible,
ABSTRACT A new equivalence notion between non-stationary subdivision schemes, termed asymptotical... more ABSTRACT A new equivalence notion between non-stationary subdivision schemes, termed asymptotical similarity, which is weaker than asymptotical equivalence, is introduced and studied. It is known that asymptotical equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotical equivalence can be relaxed to asymptotical similarity. This result applies to a wide class of non-stationary schemes of importance in theory and applications.
Convergence of exponential sums that interpolate to Laplace transforms (1.1) have been studied by... more Convergence of exponential sums that interpolate to Laplace transforms (1.1) have been studied by several authors [3, 6, 8, 15]. For rational functions that interpolate to Markov functions (also called Hamburger or Stieltjes Series or Hilbert Transforms) (1.2) far more detailed convergence results are available (see [10, 11, 16] and references therein). Both (1.1) and (1.2) are special cases of the transform (1.3) where K(x, t) is a strictly totally positive kernel.
International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, 1990
Results from the analysis of binary subdivision schemes are reviewed with a focus on properties w... more Results from the analysis of binary subdivision schemes are reviewed with a focus on properties which are special to interpolatory schemes. Some new results are presented, in particular the known sufficient conditions for the convergence of binary subdivision schemes to Cv limit functions are proved to be necessary in the case of interpolatory schemes. Specific examples of interpolatory schemes are reviewed, and their properties are concluded from the general theory.
Dyn, N. and S. Rippa, Data-dependent triangulations for scattered data interpolation and finite e... more Dyn, N. and S. Rippa, Data-dependent triangulations for scattered data interpolation and finite element approximation, Applied Numerical Mathematics 12 (1993) 89-105. In this paper we review methods for improving the accuracy of approximation by piecewise linear polynomials defined over triangulations to bivariate functions, by adapting the triangulation to the structure of the approximated function. Many criteria for optimizing triangulations as well as efficient algorithms for computing data-dependent triangulations that are locally optimal with respect to the criteria are presented. It is demonstrated that data-dependent triangulations can improve significantly the quality of approximation and that long and thin triangles, which are traditionally avoided, are sometimes necessary to have. Our results are concerned with two processes of approximation: (i) scattered data interpolation to bivariate functions, (ii) finite element approximations to solutions of second-order elliptic partial differential equations. Extensions of these ideas to higher-order approximation schemes as well as extensions to multivariate approximation problems are also discussed.
We study properties of a binary operation between two compact sets depending on a weight in [0,1]... more We study properties of a binary operation between two compact sets depending on a weight in [0,1], termed metric average. The metric average is used in spline subdivision schemes for compact sets in |R^n, instead of the Minkowski convex combination of sets, to retain non-convexity, see N. Dyn, E. Farkhi, ``Spline subdivision schemes for compact sets with metric averages", Trends in Approximation Theory (2001). Some properties of the metric average of sets in |R, like the cancellation property and the linear behavior of the Lebesgue measure of the metric average with respect to the weight, are proven. We present an algorithm for computing the metric average of two compact sets in |R, which are finite unions of intervals, as well as an algorithm for reconstructing one of the metric average's operands, given the second operand, the metric average and the weight.
Curve subdivision is a technique for generating smooth curves from initial control polygons by re... more Curve subdivision is a technique for generating smooth curves from initial control polygons by repeated refinement. The most common subdivision schemes are based on linear refinement rules, which are applied separately to each coordinate of the control points, and the analysis of these schemes is well understood. Since the resulting limit curves are not sufficiently sensitive to the geometry of the control polygons, there is a need for geometric subdivision schemes. Such schemes take the geometry of the control polygons into account by using non-linear refinement rules and are known to generate limit curves with less artefacts. Yet, only few tools exist for their analysis, because the non-linear setting is more complicated. In this paper, we derive sufficient conditions for a convergent interpolatory planar subdivision scheme to produce tangent continuous limit curves. These conditions as well as the proofs are purely geometric and do not rely on any parameterization.
Interpolatory subdivision schemes generating surfaces from initial control nets with the topology... more Interpolatory subdivision schemes generating surfaces from initial control nets with the topology of a regular grid are introduced and analyzed. The schemes are based upon the butterfly-scheme and are shown to have components with continuous first order derivatives. The generalization of these schemes for the application to general control nets is also discussed. 1 Introduction The design of surfaces in CAGD starts with a set of points with connectivity relations between them, termed control points. The simplest connectivity is that of a regular grid, namely each point has two indices: p ij , i = 1; : : : ; N 1 , j = 1; : : : ; N 2 . Thus the four points p ij ; p i+1;j ; p i;j+1 and p i+1;j+1 constitute a "face", and each point p ij has four neighbors p i;j Sigma1 , p iSigma1;j . Another possible topology of the connectivity relations is that of a triangular grid, where each face is determined by three control points and each pair of control points can belong to at most tw...
Interpolatory subdivision schemes generating surfaces from initial control nets with the topology... more Interpolatory subdivision schemes generating surfaces from initial control nets with the topology of a regular grid are introduced and analyzed. The schemes are based upon the butterfly-scheme and are shown to have components with continuous first order derivatives. The generalization of these schemes for the application to general control nets is also discussed.
Given a set of matrices, modeled as samples of a matrix-valued function, we suggest a method to a... more Given a set of matrices, modeled as samples of a matrix-valued function, we suggest a method to approximate the underline function using a product approximation operator. This operator extends known approximation methods by exploiting the structure of the matrices in the samples set, and based on decomposition theorems. We introduce our approach in detail and discuss its advantages using a few examples. In addition, we provide basic tools for analyzing properties of the matrix functions generated by our approximation operators.
Dahmen and Micchelli [8] have shown that in general the coefficients of the refined control nets ... more Dahmen and Micchelli [8] have shown that in general the coefficients of the refined control nets of a box spline surface converge to the surface at (at least) the rate of the refinement. The purpose of this article is to show that under mild additional assumptions the convergence rate is even quadratic. Although this rate is in general best possible,
ABSTRACT A new equivalence notion between non-stationary subdivision schemes, termed asymptotical... more ABSTRACT A new equivalence notion between non-stationary subdivision schemes, termed asymptotical similarity, which is weaker than asymptotical equivalence, is introduced and studied. It is known that asymptotical equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotical equivalence can be relaxed to asymptotical similarity. This result applies to a wide class of non-stationary schemes of importance in theory and applications.
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