Calibration relations for nonpolynomial splines

Nonpolynomial (X, A, ϕ)-splines of the third order and the special case of B _ϕ-splines of class C² are studied. For such splines calibration relations are obtained, owing to which the coordinate splines on the original grid is represented in terms of the coordinate splines on a refined grid. A nonlinear mapping (ℝ⁴)⁹ ↦ ℝ⁴ and locally orthogonal chains of vectors are used for this purpose. Bibliography: 22 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 39–54.     

Approximation by smoothing variational vector splines for noisy data

This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of D^m${\mathrm{D}}^m$-splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given.     

Non-uniform exponential tension splines

We describe explicitly each stage of a numerically stable algorithm for calculating with exponential tension B-splines with non-uniform choice of tension parameters. These splines are piecewisely in the kernel of D ²(D ²–p ²), where D stands for ordinary derivative, defined on arbitrary meshes, with a different choice of the tension parameter p on each interval. The algorithm provides values of the associated B-splines and their generalized and ordinary derivatives by performing positive linear combinations of positive quantities, described as lower-order exponential tension splines. We show that nothing else but the knot insertion algorithm and good approximation of a few elementary functions is needed to achieve machine accuracy. The underlying theory is that of splines based on Chebyshev canonical systems which are not smooth enough to be ECC-systems. First, by de Boor algorithm we construct exponential tension spline of class C ¹, and then we use quasi-Oslo type algorithms to evaluate classical non-uniform C ² tension exponential splines.      

Shape-preserving interpolation by splines using vector subdivision

We give a local convexity preserving interpolation scheme using parametricC ² cubic splines with uniform knots produced by a vector subdivision scheme which simultaneously provides the function and its first and second order derivatives. This is also adapted to give a scheme which is both local convexity and local monotonicity preserving when the data values are strictly increasing in thex-direction.     

On the use of splines for the numerical solution of nonlinear two-point boundary value problems

Cubic splines on splines and quintic spline interpolations are used to approximate the derivative terms in a highly accurate scheme for the numerical solution of two-point boundary value problems. The storage requirement is essentially the same as for the usual trapezoidal rule but the local accuracy is improved fromO(h ³) to eitherO(h ⁶) orO(h ⁷), whereh is the net size. The use of splines leads to solutions that reflect the smoothness of the slopes of the differential equations.     

Cone Spline Surfaces and Spatial Arc Splines – a Sphere Geometric Approach

Basic sphere geometric principles are used to analyze approximation schemes of developable surfaces with cone spline surfaces, i.e., G ¹-surfaces composed of segments of right circular cones. These approximation schemes are geometrically equivalent to the approximation of spatial curves with G ¹-arc splines, where the arcs are circles in an isotropic metric. Methods for isotropic biarcs and isotropic osculating arc splines are presented that are similar to their Euclidean counterparts. Sphere geometric methods simplify the proof that two sufficiently close osculating cones of a developable surface can be smoothly joined by a right circular cone segment. This theorem is fundamental for the construction of osculating cone spline surfaces. Finally, the analogous theorem for Euclidean osculating circular arc splines is given.     

Optimal design for smoothing splines

In the common nonparametric regression model we consider the problem of constructing optimal designs, if the unknown curve is estimated by a smoothing spline. A special basis for the space of natural splines is introduced and the local minimax property for these splines is used to derive two optimality criteria for the construction of optimal designs. The first criterion determines the design for a most precise estimation of the coefficients in the spline representation and corresponds to D-optimality, while the second criterion is the G-optimality criterion and corresponds to an accurate prediction of the curve. Several properties of the optimal designs are derived. In general, D- and G-optimal designs are not equivalent. Optimal designs are determined numerically and compared with the uniform design.     

On wavelet decomposition of Hermite type splines

We consider wavelet decompositions of spaces of Hermite type splines of class C¹(α, β) that are defined by a 4-component vector-valued function ϕ(t) ∈ C¹ (α, β) by means of a grid X (not necessarily uniform) on (α, β) ∈ ℝ¹ (the special case ϕ(t)^def = (1, t, t²,t³)^T corresponds to cubic Hermite splines). The basis wavelets obtained are compactly supported. The decomposition and reconstruction formulas are given. Bibliography: 8 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 33–46     

Univariate cubic L 1 interpolating splines based on the first derivative and on 5-point windows: analysis, algorithm and shape-preserving properties

In this paper, univariate cubic L ₁ interpolating splines based on the first derivative and on 5-point windows are introduced. Analytical results for minimizing the local spline functional on 5-point windows are presented and, based on these results, an efficient algorithm for calculating the spline coefficients is set up. It is shown that cubic L ₁ splines based on the first derivative and on 5-point windows preserve linearity of the original data and avoid extraneous oscillation. Computational examples, including comparison with first-derivative-based cubic L ₁ splines calculated by a primal affine algorithm and with second-derivative-based cubic L ₁ splines, show the advantages of the first-derivative-based cubic L ₁ splines calculated by the new algorithm.     

Gibbs-Wilbraham splines

If a function with a jump discontinuity is approximated in the norm ofL ²[–1,1] by a periodic spline of orderk with equidistant knots, a behavior analogous to the Gibbs-Wilbraham phenomenon for Fourier series occurs. A set of cardinal splines which play the role of the sine integral function of the classical phenomenon is introduced. It is then shown that ask becomes large, the phenomenon for splines approaches the classical phenomenon.Communicated by Ronald A. DeVore.