Asymptotically exact and conservative confidence bands are obtained for a nonparametric regression function, using piecewise constant and piecewise linear spline estimation, respectively. Compared to the pointwise confidence interval of Huang (2003), the confidence bands are inflated by a factor proportional to {log (n)}, with the same width order as the Nadaraya-Watson bands of Härdle (1989), ...

Abstract: In this study, the non-polynomial spline function is used to find the numerical solution of the third order singularly perturbed boundary value problems. The convergence analysis is discussed and the method is shown to have second order convergence. The order of convergence is improved up to fourth order using the improved end conditions. Numerical results are given to describe the ef...

It is well known that a cardinal B-spline of order m,m ∈ N, is a piecewise polynomial function. In this note we propose an effective method for calculating the coefficients of polynomials which constitute a cardinal B-spline. © 2010 Elsevier Ltd. All rights reserved.

C-Bézier and C-B-spline curves, as the trigonometric extensions of cubic uniform spline curves are well-known in geometric modeling. These curves depend on a shape parameter α in a way that α → 0 yields the cubic polynomial curves. The geometric effect of the alteration of this parameter is discussed in this paper by the help of relative parametrization and linear approximation.

GB-spline curves can be considered as the generalization of B-spline curve incorporating a shape parameter into the polynomial basis functions. The geometric effect of the alteration of the shape parameter is discussed in this paper, including constrained shape control of the curve.

Interpolation by various types of splines is the standard procedure in many applications. In this paper we shall discuss harmonic spline “interpolation” (on the lines of a grid) as an alternative to polynomial spline interpolation (at vertices of a grid). We will discuss some advantages and drawbacks of this approach and present the asymptotics of the Lp-error for adaptive approximation by harm...

This is a study concerning the identification of the heterogeneous flexural rigidity of a beam governed by the steady-state Euler-Bernoulli fourth order ordinary differential equation. We use the method of Variational Imbedding (MVI) to deal with the inverse problem for the coefficient identification from over-posed data. The method is identifying the coefficient by approximating it with a piec...

We present a new approach to the construction of piecewise polynomial or rational C k-spline surfaces of arbitrary topological structure. The basic idea is to use exclusively parametric smoothness conditions, and to solve the well known problems at extraordinary points by admitting singular parametrizations. The smoothness of the spline surfaces is guaranteed by specifying a regular smooth re-p...

When optimizing the performance of constrained robotic system, the motion trajectory plays a crucial role. In this research the motion planning problem for systems that admit a polynomial description of the system dynamics through differential flatness is tackled by parameterizing the system’s so-called flat output as a polynomial spline. Using basic properties of B-splines, sufficient conditio...