Improved orders of approximation derived from interpolatory cubic splines

Lets be a cubic spline, with equally spaced knots on [a, b] interpolating a given functiony at the knots. The parameters which determines are used to construct a piecewise defined polynomialP of degree four. It is shown thatP can be used to give better orders of approximation toy and its derivatives than those obtained froms. It is also shown that the known superconvergence properties of the derivatives ofs, at specific points of [a, b], are all special cases of the main result contained in the present paper.     

Bivariate interpolatory rational splines

Smoothing conditions in terms of Bézier coefficients of piecewise rational functions on an arbitrary triangulation are derived. This facilitates the solution of the problem of bivariate rational spline interpolation, with or without convexity constraints, particularly on the three and four-directional meshes. For such a triangulation, we also derive the conformality condition that a bivariate rationale spline function must satisfy, and we demonstrate the interpolation scheme with a low-degree example.The research of this author was supported by NSF Grant # DMS-92-06928.     

Overconvergence properties of quintic interpolatory splines

Let Q be a quintic spline with equi-spaced knots on [a, b] interpolating a given function y at the knots. The parameters which determine Q are used to construct a piecewise defined polynomial P of degree six. It is shown that P can be used to give at any point of [a, b] better orders of approximation to y and its derivatives than those obtained from Q. It is also shown that the superconvergence properties of the derivatives of Q, at specific points of [a, b], are all simple consequences of the properties of P.     

Computation of interpolatory splines via triadic subdivision

We present an algorithm for the computation of interpolatory splines of arbitrary order at triadic rational points. The algorithm is based on triadic subdivision of splines. Explicit expressions for the subdivision symbols are established. These are rational functions. The computations are implemented by recursive filtering.     

On approximation of functions from below by splines of the best approximation with free nodes

Let M be the set of functions integrable to the power β=(r+1+1/p)^-1. We obtain asymptotically exact lower bounds for the approximation of individual functions from the set M by splines of the best approximation of degree rand defect k in the metric of L _p.     

On the best approximation of periodic functions of two variables by polynomial splines

We consider the problem of the best approximation of periodic functions of two variables by a subspace of splines of minimal defect with respect to a uniform partition.     

Interpolation by quadratic splines

Convergence properties of quadratic spline interpolation of continuous functions that does not necessarily take place at the midpoints of mesh intervals are investigated. A theorem giving lower bounds on the elements of the inverse of certain tridiagonal matrices is proved. This result is used to precisely relate the norm of certain interpolating projections to the points of interpolation and local mesh ratios. It is shown, for example, that for Lipschitz continuous functions, any choice of interpolation points, one in each mesh interval, uniformly bounded away from the mesh points, yields convergence at the best possible rate with no mesh ratio restriction.     

On the best approximation and polynomials of the least quadratic deviation

Analysis Mathematica - ПустьС 2π — пространс тво...     

Upper bounds for the best one-sided approximation by splines of the classes WrL1

In the present note we will investigate the problem of the one-sided approximation of functions by n-dimensional subspaces. In particular, we will find the exact value of the best one-sided approximation of the class W^rL₁ (r=1, 2, ...) of all periodic functions f(x) of period 2 for which f^(r–1)(x) (f⁽⁰⁾(x)=f(x)) is absolutely continuous and f^(r)L₁1 by periodic spline functions S_2n ( = 0, 1, ..., n=1, 2, ...) of period 2, order ,and deficiency 1.Translated from Matematicheskie Zametki, Vol. 19, No. 1, pp. 11–17, January, 1976.     

The order of the best one-sided approximation by polynomials and splines in the Lp-metric

In this paper we determine the order of the best one-sided approximation by polynomials and splines of minimal defect of the classes W^rL_p in the L_p-metric.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 323–329, March, 1976.     

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.     

On the invariant translation approximation property for discrete groups

Recently J.Roe considered the question of whether for a discrete group the reduced group -algebra is the fixed point algebra of Ad acting on the uniform Roe algebra . is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact , is equivalent to the operator space approximation property of . We also give a new characterization of exactness and a short proof of the equivalence of exactness of and exactness of for discrete groups.

     

A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines

This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes.