Fitting data using optimal Hermite type cubic interpolating splines

In this work we obtain a new optimal property for cubic interpolating splines of Hermite type applied to data-fitting problems. The existence and uniqueness of the Hermite type cubic spline with minimal quadratic oscillation in average are proved.     

A necessary condition for convergence of interpolating parabolic and cubic splines

Let the sequence of nets _n be such that , where h_i ⁽ⁿ⁾ are the lengths of the segments of a net. The bound is necessary in order that interpolating parabolic and cubic splines converge for any function in C ( = 0) or C(0 < < 1), where C is the class of functions satisfying a Lipschitz condition of order. It is also shown that this bound cannot essentially be weakened.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 165–178, February, 1976.The author thanks Yu. N. Subbotin for a useful discussion of the results obtained.     

End conditions for interpolatory cubic splines with unequally spaced knots

A class of end conditions is derived for cubic spline interpolation at unequally spaced knots. These conditions are in terms of function values at the knots and lead to 0 (h⁴) convergence uniformly on the interval of interpolation.     

Numerical calculation of fourier integrals with cubic splines

The most used formula for calculation of Fourier integrals is Filon's formula which is based on approximation of the function by a quadratic in each double interval. In order to obtain a better approximation we use the cubic spline fit. The method is not restricted to equidistant points, but the final formulas are only derived in this case. Test computations show that the spline formula may be superior to Filon's formula.     

Approximation by interpolating variational splines

In this paper we present an approximation problem of parametric curves and surfaces from a Lagrange or Hermite data set. In particular, we study an interpolation problem by minimizing some functional on a Sobolev space that produces the new notion of interpolating variational spline. We carefully establish a convergence result. Some specific cases illustrate the generality of this work.     

Convex interpolating splines of arbitrary degree II

For given data {(x _i ,y _i )} _i=0 ⁿ , (x ₀<x ₁<...<x _n ) we consider the possibility of finding a spline functions of arbitrary degreek+1 (k 1) with preassigned smoothnessl, where 1 l [(k+1)/2]. The splines should be such thats(x _i )=y _i ,i=0, 1,...,n ands is increasing and convex on [x ₀,x _n ]. Sufficient conditions which guarantee the existence ofs and an explicit formula for this function are derived.     

Local exponential splines with arbitrary knots

We construct local L-splines that have an arbitrary arrangement of knots and preserve the kernel of a linear differential operator L of order r with constant coefficients and real pairwise distinct roots of the characteristic polynomial.     

A note on polynomial splines with free knots

It is observed that the tangent spaces to sets of splines with free knots can often be characterized as spaces of splines with fixed knots. It follows that some recent theorems on nonlinear approximation are applicable in this setting.This work was supported in part by the National Science Foundation.     

Uniform approximation by some Hermite interpolating splines

In this paper some upper bound for the error ∥ s-f ∥_∞ is given, where f ε C¹[a,b], but s is a so-called Hermite spline interpolant (HSI) of degree 2q ?1 such that f(x_i) = s(x_i), f′(rmx_i) = s′(x_i), s^(j) (x_i) = 0 (i = 0, 1, …, n; j = 2, 3, …, q ?1; n > 0, q > 0) and the knots x_i are such that a = x₀ < x₁ < … < x_n = b. Necessary and sufficient conditions for the existence of convex HSI are given and upper error bound for approximation of the function fε C¹[a, b] by convex HSI is also given.     

Smooth,easy to compute interpolating splines

We present a system of interpolating splines with first-order and approximate second-order geometric continuity. The curves are easily computed in linear time by solving a diagonally dominant, tridiagonal system of linear equations. Emphasis is placed on the need to find aesthetically pleasing curves in a wide range of circumstances; favorable results are obtained even when the knots are very unequally spaced or widely separated. The curves are invariant under translation, rotation, and scaling, and the effects of a local change fall off exponentially as one moves away from the disturbed knot.Approximate second-order continuity is achieved by using a linear mock curvature function in place of the actual endpoint curvature for each spline segment and choosing tangent directions at knots so as to equalize these. This avoids extraneous solutions and other forms of undesirable behavior without seriously compromising the quality of the results.The actual spline segments can come from any family of curves whose endpoint curvatures can be suitably approximated, but we propose a specific family of parametric cubics. There is freedom to allow tangent directions and tension parameters to be specified at knots, and special curl parameters may be given for additional control near the endpoints of open curves.This research was supported in part by the National Science Foundation under grants IST-820-1926 and MCS-83-00984 and by the Systems Development Foundation.     

Discrete weighted cubic splines

This paper presents methods for shape preserving spline interpolation. These methods are based on discrete weighted cubic splines. The analysis results in two algorithms with automatic selection of the shape control parameters: one to preserve the data monotonicity and other to retain the data convexity. Discrete weighted cubic B-splines and control point approximation are also considered.     

Optimal smoothing and interpolating splines with constraints

This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approximation of discontinuous functions, and trajectory planning.     

Convex interpolating splines of arbitrary degree. III

For given data (x _i, f_i) _i=0 ⁿ (x ₀<x ₁<...<x _n) we consider the possibility of finding a spline functions of arbitrary degreek (k3) with preassigned smoothnessl, where 1l[(k-1)/2]. The splines should be such thats(x _i)=f _i (i=0, 1,...,n) ands is convex or nondecreasing and convex on [x ₀,x _n]. An explicit formula for this function as well as the conditions that guarantee the required properties are established. An algorithm for the determination of the splines and the error bounds is also included.     

Convergence of interpolating cardinal splines: Power growth

Letf(x) be the restriction to the real axis of an entire function of exponential typeτ<π and of power growth on the axis. Then thenth order cardinal spline,ℒ _nf(x), interpolatingf(x) at the integers converges uniformly on compacta tof(x). This is also true of the respective derivatives. An example shows that exponential typeπ is not necessarily permitted. The proof utilizes distribution theory and estimates on the derivatives of the Fourier transform of the fundamental splineL _n(x). This research is partially supported by Canadian National Research Council Grant A-7687.     

Interpolation by periodic splines with Birkhoff knots

Summary We give a complete characterization of the Hermite interpolation problem by periodic splines with Birkhoff knots. As a dual result we derive the characterization of the Birkhoff interpolation by periodic splines with multiple knots.Sponsored by the Bulgarian Ministry of Education and Science under Contract No. MM-15     

Constrained approximation by splines with free knots

In this paper, a method that combines shape preservation and least squares approximation by splines with free knots is developed. Besides the coefficients of the spline a subset of the knot sequence, the so-calledfree knots, is included in the optimization process resulting in a nonlinear least squares problem in both the coefficients and the knots. The original problem, a special case of aconstrained semi-linear least squares problem, is reduced to a problem that has only the knots of the spline as variables. The reduced problem is solved by a generalized Gauss-Newton method. Special emphasise is given to the efficient computation of the residual function and its Jacobian. Dedicated to our colleague and teacher Prof. Dr. J. W. Schmidt on the occasion of his 65th birthday Research of the first author was supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1,2-2.     

Bivariate interpolating polynomials and splines (I)

The multivariate splines which were first presented by de Boor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development. The author of this paper is interested in the area of interpolation with special emphasis on the interpolation methods and their approximation orders. But such B-splines (both univariate and multivariate) do not interpolated directly, so I approached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case (See[7]) to multivariate case. I selected triangulated region which is inspired by other mathematician’s works (e.g. [2] and [3]) and extend the interpolating polynomials from univariate to m-variate case (See [10])In this paper some results in the case m=2 are discussed and proved in more concrete details. Based on these polynomials, the interpolating splines (it is defined by me as piecewise polynomials in which the unknown partial derivatives are determined under certain continuous conditions) are also discussed. The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated. We limited our discussion on the rectangular domain which is partitioned into equal right triangles. As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains, we will discuss in the next paper.     

Shape-preserving interpolation by cubic splines

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.     

Weighted cubic and biharmonic splines

In this paper we discuss the design of algorithms for interpolating discrete data by using weighted cubic and biharmonic splines in such a way that the monotonicity and convexity of the data are preserved. We formulate the problem as a differential multipoint boundary value problem and consider its finite-difference approximation. Two algorithms for automatic selection of shape control parameters (weights) are presented. For weighted biharmonic splines the resulting system of linear equations can be efficiently solved by combining Gaussian elimination with successive over-relaxation method or finite-difference schemes in fractional steps. We consider basic computational aspects and illustrate main features of this original approach.