New upper and lower bounds for the L2 and Vo norms of derivatives of the error in polynomial spline interpolation are derived. These results improve corresponding results of Ahlberg, Nilson, and Walsh, cf. [1], and Schultz and Varga, cf. [5].

This paper explores numerical solutions of optimal control problems using B–Spline curves. It is aimed to give a general framework on how to use B–Splines to formulate optimal control problems and to solve them numerically using Nonlinear Trajectory Generation software package. Effects of the selection of the B–Spline parameters, such as, number of intervals, smoothness, piecewise polynomial or...

Multivariate spline functions are smooth piecewise polynomial functions over triangulations consisting of n-simplices in the Euclidean space IR. A straightforward method for using these spline functions to fit given scattered data and numerically solve elliptic partial differential equations is presented . This method does not require constructing macro-elements or locally supported basis funct...

We analyze an alternative to classical Zernike fitting based on the cubic B-spline model, and compare the strengths and weaknesses of each representation over a set of different wavefronts that cover a wide range of shape complexity. The results obtained show that a Zernike low-degree polynomial expansion or a cubic B-spline with a low number of breakpoints are the best choices for fitting simp...

We generalize the univariate divided difference to a multivariate setting by considering linear combinations of point evaluations that annihilate the null space of certain differential operators. The relationship between such a linear functional and polynomial interpolation resembles that between the divided difference and Lagrange interpolation. Applying the functional to the shifted multivari...

A sphere is a fundamental geometric object widely used in (computer aided) design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal approximation of equilateral spherical triangles by patches if measure quality (simplified) radial error. As consequence, approximations unit spline underlying on par...

This paper points out that the fundamental partial differential equation (PDE) implies that the drifts of the underlying processes are irrelevant, regardless whether the state factors are tradable or not. Since the simple boundary condition for default-free discount bonds can be satisfied by a linear discount function, the variances and covariances of the underlying processes are also irrelevan...

Conccpts and techniques for thc construction of spline surfaces over irregular meshes are developed and made concrete by exhibiting a family of 0 2 surface splines. These splines extend the B-spline paradigm for the construction of parametric piecewise polynomial surfaces to control_meshe9 with non quadrila.leral cells and mOre or fewer Lhan four cells meeting at a point. In particular, the mes...

The matrix forms for curves and surfaces were largely promoted in CAD/CAM. In this paper we have presented two matrix representation formulations for arbitrary degree NURBS curves and surfaces explicitly other than recursively. The two approaches are derived from the computation of divided difference and the Marsden identity respectively. The explicit coefficient matrix of B-spline with equally...