Comonotone Adaptive Interpolating Splines

For any given data we propose the construction of an interpolating spline of class C ¹, which is either a quadratic polynomial or a linear/linear rational function between the knots, and preserves the monotonicity of the data on the sections of rational intervals. We prove the uniqueness and existence of this spline. Numerical tests show good approximation properties and flexibility due to the non-coincidence of the given data arguments and the spline knots which can be chosen freely.     

Behaviour of Exponential Splines as Tensions Increase without Bound

Schweikert (J. Math. Phys.45(1966), 312–317) showed that for sufficiently high tensions an exponential spline would have no more changes in sign of its second derivative than there were changes in the sign of successive second differences of its knot sequence. Späth (Computing4(1969), 225–233) proved the analogous result for first derivatives, assuming uniform tension throughout the spline. Later, Pruess (J. Approx. Theory17(1976), 86–96) extended Späth's result to the case where the inter-knot tensionsp_imay not all be the same but tend to infinity at the same asymptotic growth rate, in the sense thatp_i∈Θ(p₁) for alli. This paper extends Pruess's result by showing his hypothesis of uniform boundedness of the tensions to be unnecessary. A corollary is the fact that for high enough minimum interknot tension, the exponential spline through monotone knots will be a ²monotone curve. In addition, qualitative bounds on the difference in slopes between the interpolating polygon and the exponential spline are developed, which show that Gibbs-like behaviour of the spline's derivative cannot occur in the neighbourhood of the knots.     

Explicit Error Bounds for Periodic Splines of Odd Order on a Uniform Mesh

The dependence relationships connecting equal interval splinesand their derivatives are analysed to obtain the form of theerror term when the spline is replaced by a general function.The defining equations for periodic splines of odd order ona uniform mesh are then expressed in terms of a positive definitecirculant matrix A and attainable bounds determined for thecondition number of A and for the norm of A^-1. In conjunctionwith the error term associated with the dependence relationships,this enables explicit error bounds to be established for thederivatives at the knots of the spline function. Some subsidiary results in the paper also relate to B-splineson a uniform mesh.     

Computing ECT-B-splines recursively

ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval [a,b] connected at inner knots all of multiplicities zero by full connection matrices A ^[i] that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves. *Supported in part by INTAS 03-51-6637.     

Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form

The problems of determining the B–spline form of a C ² Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C ² quintic bases on uniform triple knots are constructed for both open and closed C ² curves, and are used to derive simple explicit formulae for the B–spline control points of C ² PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C ² to C ¹. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.     

On diagrammatic bounds of knot volumes and spectral invariants

In recent years, several families of hyperbolic knots have been shown to have both volume and λ₁ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or λ₁. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on λ₁. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded.     

On \pi-hyperbolic knots with the same 2-fold branched coverings

We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in the set of knots under the equivalence relation if is homeomorphic to . It is known that there exist arbitrarily many different hyperbolic knots with the same 2-fold branched coverings, due to mutation along Conway spheres. Thus the most basic class of knots to investigate are knots which do not admit Conway spheres. In this paper we solve the above problem for knots which do not admit Conway spheres, in the following sense: we give upper bounds for the number of knots in the equivalence class [K] of a knot K and we describe how the different knots in the equivalence class of K are related. Received: 3 August 1998 / in final form: 17 June 1999     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.     

Cubic Splines with Minimal Norm

Natural cubic interpolatory splines are known to have a minimal L ₂-norm of its second derivative on the C ² (or W ₂ ² ) class of interpolants. We consider cubic splines which minimize some other norms (or functionals) on the class of interpolatory cubic splines only. The cases of classical cubic splines with defect one (interpolation of function values) and of Hermite C ¹ splines (interpolation of function values and first derivatives) with spline knots different from the points of interpolation are discussed.     

The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions

The BS Hermite spline quasi-interpolation scheme is presented. It is related to the continuous extension of the BS linear multistep methods, a class of Boundary Value Methods for the solution of Ordinary Differential Equations. In the ODE context, using the numerical solution and the associated numerical derivative produced by the BS methods, it is possible to compute, with a local approach, a suitable spline with knots at the mesh points collocating the differential equation at the knots and having the same convergence order as the numerical solution. Starting from this spline, here we derive a new quasi-interpolation scheme having the function and the derivative values at the knots as input data. When the knot distribution is uniform or the degree is low, explicit formulas can be given for the coefficients of the new quasi-interpolant in the B-spline basis. In the general case these coefficients are obtained as solution of suitable local linear systems of size 2d×2d, where d is the degree of the spline. The approximation order of the presented scheme is optimal and the numerical results prove that its performances can be very good, in particular when suitable knot distributions are used.     

On the minimum ropelength of knots and links

The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are C ^1,1 curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp. Oblatum 11-IV-2001 & 20-III-2002?Published online: 17 June 2002     

Detection of C2‐singularities using uniform spline approximations

For the detection of C²‐singularities, we present lower estimates for the error in Schoenberg variation‐diminishing spline approximation with equidistant knots in terms of the classical second‐order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second‐order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.     

Convergence of Derivatives of Optimal Nodal Splines

Optimal nodal spline interpolantsWfof ordermwhich have local support can be used to interpolate a continuous functionfat a set of mesh points. These splines belong to a spline space with simple knots at the mesh points as well as atm−2 arbitrary points between any two mesh points and they reproduce polynomials of orderm. It has been shown that, for a sequence of locally uniform meshes, these splines converge uniformly for anyfCas the mesh norm tends to zero. In this paper, we derive a set of sufficient conditions on the sequence of meshes for the uniform convergence ofD^jWftoD^jfforfC^sandj=1, …, s<m. In addition we give a bound forD^rWfwiths<r<m. Finally, we use optimal nodal spline interpolants for the numerical evaluation of Cauchy principal value integrals.     

On Optimal Error Bounds for Derivatives of Interpolating Splines on a Uniform Partition

Based on Peano kernel technique, explicit error bounds (optimal for the highest order derivative) are proved for the derivatives of cardinal spline interpolation (interpolating at the knots for odd degree splines and at the midpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigation of their zero distributions. The bounds are given in terms of Euler–Frobenius polynomials and their zeros.     

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.     

Cobordism of disk knots

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots D ⁿ⁻² ↪ D ⁿ . Any two disk knots are cobordant if the cobordisms are not required to fix the boundary sphere knots, and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of J. Levine on the cobordism of sphere knots, we define disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cobordant rel boundary if and only if their disk knot Seifert matrices are algebraically cobordant. We also ask which algebraic cobordism classes can be realized given a fixed boundary knot and provide a complete classification when the boundary knot has no 2-torsion in its middle-dimensional Alexander module. In the course of this classification, we establish a close connection between the Blanchfield pairing of a disk knot and the Farber-Levine torsion pairing of its boundary knot (in fact, for disk knots satisfying certain connectivity assumptions, the disk knot Blanchfield pairing will determine the boundary Farber-Levine pairing). In addition, we study the dependence of disk knot Seifert matrices on choices of Seifert surface, demonstrating that all such Seifert matrices are rationally S-equivalent, but not necessarily integrally S-equivalent.     

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.     

Exact Markov-Type Inequalities for Oscillating Perfect Splines

We prove the inequality for each perfect spline s of degree r with n-r knots and n zeros in [-1,1] . Here T _rn is the Tchebycheff perfect spline of degree r with n-r knots, normalized by the condition ||T _rn || _C[-1,1] := max  _{-1≤ x ≤ 1} |T _rn (x)| =1 . The constant ||T _rn ^(k) || _Lp[-1,1] in the above inequality is the best possible. June 8, 1999. Date revised: May 31, 2000. Date accepted: January 16, 2001.     

Natural spline functions,their associated eigenvalue problem

Summary We consider the problem of studying the behaviour of the eigenvalues associated with spline functions with equally spaced knots. We show that they are wherem is the order of the spline andn, the number of knots.This result is of particular interest to prove optimality properties of the Generalized Cross-Validation Method and had been conjectured by Craven and Wahba in a recent paper.     

On band circulant matrices in the periodic spline interpolation theory

We review and extend the analysis of band circulant matrices which occur in the periodic spline interpolation theory with equispaced knots. Explicit bounds on the inverse matrices are given.