On the approximation power of generalized T-splines (original) (raw)
Spaces of generalized splines over T-meshes
Journal of Computational and Applied Mathematics, 2015
We consider a class of non-polynomial spline spaces over T-meshes, that is, of spaces locally spanned both by polynomial and by suitably-chosen non-polynomial functions, which we will refer to as generalized splines over T-meshes. For such spaces, we provide, under certain conditions, a dimension formula and a basis based on the notion of minimal determining set. We explicitly examine some relevant cases, which enjoy a noteworthy behaviour with respect to differentiation and integration; finally, we also study the approximation power of the just constructed spline spaces.
On two families of near-best spline quasi-interpolants on non-uniform partitions of the real line
Arxiv preprint math/ …, 2006
The univariate spline quasi-interpolants (abbr. QIs) studied in this paper are approximation operators using B-spline expansions with coefficients which are linear combinations of discrete or weighted mean values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find QIs which have both good approximation orders and uniform norms which are bounded independently of the given partition. Near-best QIs are obtained by minimizing an upper bound of the infinity norm of QIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of two families of near-best QIs of approximation order 3.
Optimal local spline interpolants
Journal of Computational and Applied Mathematics, 1987
A constructive proof is given of the existence of a local spline interpolant which also approximates optimally in the sense that its associated operator reproduces polynomials of maximal order. First, it is shown that such an interpolant does not exist for orders higher than the linear case if the partition points of the appropriate spline space coincide with the given interpolation points. Next, in the main result, the desired existence of an optimal local spline interpolant for all orders is proved by increasing, in a specified manner, the set of partition points. Although our interpolant reproduces a more restricted function space than its quasi-interpolant counterpart constructed by De Boor and Fix [l], it has the advantage of interpolating every real function at a given set of points. Finally, we do some explicit calculations in the quadratic case.
Boletim da Sociedade Paranaense de Matemática, 2021
In this paper, we use the finite element method to construct a new normalized basis of a univariate quadratic C1C^1C1 spline space. We give a new representation of Hermite interpolant of any piecewise polynomial of class at least C1C^1C1 in terms of its polar form. We use this representation for constructing several superconvergent and super-superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.
A general method for constructing quasi-interpolants from B-splines
Journal of Computational and Applied Mathematics, 2010
A general method for constructing quasi-interpolation operators based on B-splines is developed. Given a B-spline φ in R s , s ≥ 1, normalized by i∈Z s φ (• − i) = 1, the classical structure Q (f) := i∈Z s λf (• + i) φ (• − i), for a quasi-interpolation operator Q is considered. A minimization problem is derived from an estimate of the quasi-interpolation error associated with Q when λf is a linear combination of values of f at points in some neighbourhood of the support of φ; or a linear combination of values of f and some of its derivatives at some points in this set; or a linear combination of weighted mean values of the function to be approximated. That linear functional is defined to produce a quasiinterpolant exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of φ. The solution of that minimization problem is characterized in terms of specific splines which do not depend on λ but only on φ.
On Some Generalizations of B-Splines
2019
In this article, we consider some generalizations of polynomial and exponential B-splines. Firstly, the extension from integral to complex orders is reviewed and presented. The second generalization involves the construction of uncountable families of self-referential or fractal functions from polynomial and exponential B-splines of integral and complex orders. As the support of the latter B-splines is the set [0, ∞), the known fractal interpolation techniques are extended in order to include this setting.
Mixed interpolating–smoothing splines and the ν-spline
Journal of Mathematical Analysis and Applications, 2006
In their monograph, Bezhaev and Vasilenko have characterized the "mixed interpolatingsmoothing spline" in the abstract setting of a Hilbert space. In this paper, we derive a similar characterization under slightly more general conditions. This is specialized to the finite-dimensional case, and applied to a few well-known problems, including the ν-spline (a piecewise polynomial spline in tension) and near-interpolation, as well as interpolation and smoothing. In particular, one of the main objectives in this paper is to show that the ν-spline is actually a mixed spline, an observation that we believe was not known prior to this work. We also show that the ν-spline is a limiting case of smoothing splines as certain weights increase to infinity, and a limiting case of near-interpolants as certain tolerances decrease to zero. We conclude with an iteration used to construct curvaturebounded ν-spline curves.