The purpose of this note is to study the Euclidean condition number of the matrix resulting from using the well-known RayleighRitz-Galerkin method with finite dimensional subspaces of polynomial spline functions to approximate the solution of a linear, selfadjoint* two-point boundary value problem. Roughly speaking, we consider a model class of such problems of order In and determine upper boun...

This paper provides a survey of shape parameterization techniques for multidisciplinary optimization and highlights some emerging ideas. The survey focuses on the suitability of available techniques for complex conngurations, with suitability criteria based on the ee-ciency, eeectiveness, ease of implementation, and availability of analytical sensitivities for geometry and grids. The paper also...

We consider polynomial spline spaces S r d (4) of degree d and smoothness r deened on triangulations. It is known that for d 3r + 2, S r d (4) possesses a basis of star-supported splines, i.e., splines whose supports are at most the set of triangles surrounding a vertex. Here we extend the theory by showing that for all d 3r + 1, there exist triangulations for which no such bases exist. 1. Intr...

We prove that for a 3-monotone function F ∈ C[−1, 1], one can achieve the pointwise estimates |F(x) − Ψ(x)| ≤ cω3(F, ρn(x)), x ∈ [−1, 1], where ρn(x) := 1 n2 + √ 1−x2 n and c is an absolute constant, both with Ψ , a 3-monotone quadratic spline on the nth Chebyshev partition, and with Ψ , a 3-monotone polynomial of degree ≤ n. The basis for the construction of these splines and polynomials is th...

Article history: Available online 30 June 2010

This paper investigates the mathematical background of multiresolution analysis in the specific context where the signal is represented by irregularly sampled data at known locations. The study is related to the construction of nested piecewise polynomial multiresolution spaces represented by their corresponding orthonormal bases. Using simple spline basis orthonormalization procedures involves...

Abstract To obtain higher order continuous tool path, a cubic B-spline curve fitting method is presented in this paper to process the linear path. First, local polynomial with bidirectional scanning scheme proposed determine some initial dominant points. Based on these points, knot vector generated by average method, and constructed. constrain error, Newton’s based error estimation point insert...

We consider ®rst the spline smoothing nonparametric estimation with variable smoothing parameter and arbitrary design density function and show that the corresponding equivalent kernel can be approximated by the Green function of a certain linear differential operator. Furthermore, we propose to use the standard (in applied mathematics and engineering) method for asymptotic solution of linear d...

We present a simple, efficient, and stable method for computing—with any desired precision—the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the a...

We construct a suitable B-spline representation for a family of bivariate spline functions with smoothness r ≥ 1 and polynomial degree 3r − 1. They are defined on a triangulation with PowellSabin refinement. The basis functions have a local support, they are nonnegative and they form a partition of unity. The construction involves the determination of triangles that must contain a specific set ...