Pruess's investigations [J. Approx. Theory, 17 (1976), pp. 86–96], [Math. Comp., 33 (1979), pp. 1273–128 1] revealed the shape preservation properties of exponential splines and provided the impetus for further theoretical study of exponential splines [J. Approx. Theory, to appear]. Together, these theoretical results form the backdrop for the detailed analysis of issues in the computation of exponential splines contained herewith. Specifically, first and foremost the construction of tension parameter selection algorithms is considered. The conditioning and iterative solution of the spline equations, as well as the derivation and accuracy of end conditions, are discussed. This inquiry concludes with a potpourri of numerical considerations and the presentation of a variety of numerical examples.

Computation of Exponential Splines.

Linear Oprators And Approximation Theory

We offer the algorithm for choosing tension parameters of the generalized splines for convexity preserving interpolation. The resulting spline minimally differs from the classical cubic spline and coincides with it if sufficient convexity conditions are satisfied for the last one. We consider specific algorithms for different generalized cubic splines such as rational, exponential, variable power, hyperbolic splines, and splines with additional knots.

Near-optimal tension parameters in convexity preserving interpolation by generalized cubic splines

We consider the problem on shape-preserving interpolation by classical cubic splines. Namely, we consider conditions guaranteeing that, for a positive function (or a function whose kth derivative is positive), the cubic spline (respectively, its kth derivative) is positive. We present a survey of known results, completely describe the cases in which boundary conditions are formulated in terms of the first derivative, and obtain similar results for the second derivative. We discuss in detail mathematical methods for obtaining sufficient conditions for shape-preserving interpolation. We also develop such methods, which allows us to obtain general conditions for a spline and its derivative to be positive. We prove that, for a strictly positive function (or a function whose derivative is positive), it is possible to find an interpolant of the same sign as the initial function (respectively, its derivative) by thickening the mesh.

Shape-Preservation Conditions for Cubic Spline Interpolation

We consider the problem of approximating a function defined on a uniform mesh by the method of local polynomial spline-approximation where the mesh of the nodes of the spline is chosen displaced relative to the mesh of the initial data. Conditions are established for the local form preservation by the spline of the initial data.We study the approximative properties of the method for the case of the simplest local approximation formula and find the optimal values of the displacement parameters.

/pdf/local-approximation-by-splines-with-displacement-of-nodes-52vhqrwd2z.pdf

Local approximation by splines with displacement of nodes

We consider the problem of shape-preserving interpolation by cubic splines. We propose a unified approach to the derivation of sufficient conditions for the k-monotonicity of splines (the preservation of the sign of any derivative) in interpolation of k-monotone data for k = 0, …, 4.

Shape-Preserving Interpolation by Cubic Splines

A method for solving the inverse kinematic problem of determining the velocity characteristic of a medium from a vertical seismic survey, is proposed. It is based on the combined use of the eikonal equation and spline methods of approximation for multivariable functions. The problem is solved by assuming a horizontally stratified medium; no assumptions about the number of layers and their thickness are made. First, using the data of the first arrival times of the seismic signal from several shotpoints, which are registered by detectors located in the vertical borehole, a spline approximating the function of first arrival time of the signal from source points to any point in the Earth subsurface is constructed. Then with the help of the eikonal equation, the characteristic of the medium around the borehole is determined. Numerical experiments on the model and the real data show high efficiency of the proposed method.

http://www.ipgg.sbras.ru/ru/files/publications/ibc/cejm-2013-11-4-779.pdf

Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey

/pdf/formosokhraniaiushchaia-interpoliatsiia-kubicheskimi-41javzlngq.pdf

Формосохраняющая интерполяция кубическими сплайнами@@@Shape-Preserving Interpolation by Cubic Splines

We consider the so-called simplest formula for local approximation by polynomial splines of order 
 $ n $
 (Schoenberg splines).
The spline itself and all derivatives except that of the highest order,
approximate a given function and its corresponding derivatives with the second order.
We show that the jump of the highest derivative of order $ n-1 $
 ; i. e., the value of discontinuity, divided by the meshsize,
approximates the $ n $
 th derivative of the original function.
We found an asymptotic expansion of the jump.

V. V. Bogdanov

Papers

Shape-Preserving Interpolation by Cubic Splines

Near-optimal tension parameters in convexity preserving interpolation by generalized cubic splines

Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey

Формосохраняющая интерполяция кубическими сплайнами@@@Shape-Preserving Interpolation by Cubic Splines

On Error Estimates of Local Approximation by Splines