The paper studies quasi interpolation by scaled shifts of a smooth and rapidly decaying function. The centers are images of a smooth mapping of the hZn lattice in Rs, s n, and the scaling parameters are proportional to h. We show that for a large class of generating functions the quasi interpolants provide high order approximations up to some prescribed accuracy. Although the approximants do no...

As a rule, an energy method is widely adopted for b-spline curve smoothing, but this method has the disadvantages such as massive calculation, computation complexity and low efficiency. Compared with the energy method, multi-resolution smoothing approaches nicely overcome these obstacles. Presently, some researches have been conducted on multi-resolution smoothing, but these efforts mainly aime...

Spline methods provide an important tool to solve singularly perturbed boundary value problems. The present paper gives a comprehensive review of computational methods based on splines used in the solution of various classes of singularly perturbed problems such as self adjoint, linear, non-linear, semi-linear, quasi-linear, singular, single parameter and multi parameter. The spline based numer...

We prove an identity for basis functions of a general family of positive linear operators. It covers as special cases the Bernstein, Szász–Mirakjan and Baskakov operators. A corollary of our result can be considered a pointwise orthogonality relation. The Bernstein case is the univariate case of a remarkable identity which recently was presented by Jetter and Stöckler. As an application we give...

We investigate the construction of local quasi-interpolation schemes for a family of bivariate spline functions with smoothness r ≥ 1 and polynomial degree 3r−1. These splines are defined on triangulations with Powell-Sabin refinement, and they can be represented in terms of locally supported basis functions which form a convex partition of unity. Using the blossoming technique, we first derive...

In this paper we consider quasi-interpolatory spline operators that satisfy some interpolation conditions. We give some applications of these operators constructing approximating integral operators and numerically solving Volterra integral equations of the second kind. We prove convergence results for the constructed methods and we perform numerical examples and comparisons with other spline me...

Abstract. In this paper, the authors present a method to construct a smooth B-spline curve which fairly fits 3D points and at the same time satisfies the length constraints. By using the Lagrange multiplier’s conditional extreme, resorting to Broyden method, and then finding the least square solution of the control points, we can obtain the fair quasi-fitting modeling B-spline curve.

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of r...

Continuous set-valued functions with convex images can be approximated by known positive operators of approximation, such as the Bernstein polynomial operators and the Schoenberg spline operators, with the usual sum between numbers replaced by the Minkowski sum of sets. Yet these operators fail to approximate set-valued functions with general sets as images. The Bernstein operators with growing...