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.      

On spline Galerkin methods for singular integral equations with piecewise continuous coefficients

Summary This article analizes the convergence of the Galerkin method with polynomial splines on arbitrary meshes for systems of singular integral equations with piecewise continuous coefficients inL ² on closed or open Ljapunov curves. It is proved that this method converges if and, for scalar equations and equidistant partitions, only if the integral operator is strongly elliptic (in some generalized sense). Using the complete asymptotics of the solution, we provide error estimates for equidistant and for special nonuni-form partitions.     

On the Interpolation by Discrete Splines with Equidistant Nodes

In this paper we consider equidistant discrete splines S(j), j , which may grow as O(|j|^s) as |j|→∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of the discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of the sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order.     

Quantum B-splines

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q (quantum)-blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden’s identity.     

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.     

Fair upper bounds for the curvature in univariate convex interpolation

In convex interpolation the curvature of the interpolants should be as small as possible. We attack this problem by treating interpolation subject to bounds on the curvature. In view of the concexity the lower bound is equal to zero while the upper bound is assumed to be piecewise constant. The upper bounds are called fair with respect to a function class if the interpolation problem becomes solvable for all data sets in strictly convex position. We derive fair a priori bounds for classes of quadraticC ¹, cubicC ², and quarticC ³ splines on refined grids.     

Elementarym-harmonic cardinal B-splines

We generalize the notion of B-spline to the thin plate splines and to otherd-dimensional polyharmonic splines as defined in [Duchon, [3]]; for regular nets, we give the main properties of these B-splines: Fourier transform, decay when x , stability, integration property, links between B-splines of different orders or of different dimensions and in particular link with the polynomial B-splines, approximation using B-splines... We show that, in some sense, B-splines may be considered as a regularized form of the Dirac distribution.     

Cubic Spline Prewavelets on the Four-Directional Mesh

Dedicated to Professor M. J. D. Powell on the occasion of his sixty-fifth birthday and his retirement. In this paper, we design differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact support. They are given in closed form, and provide stable, orthogonal decompositions of L ² (R ² ) . In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting to a box spline of higher polynomial degree.     

Galerkin methods with splines for singular integral equations over (0, 1)

Summary In this paper a convergence analysis of Galerkin methods with splines for strongly elliptic singular integral equations over the interval (0, 1) is given. As trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions, multiplied by a weight function. Using inequalities of Gårding type for singular integral operators in weightedL ² spaces and the complete asymptotics of solutions at the endpoints, we provide error estimates in certain Sobolev norms.     

Approximation by cubicC 1-splines on arbitrary triangulations

Summary In this paper, we present an efficient representation for bivariate piecewise cubicC ¹-splines on arbitrary triangulations. A numerical method is discussed for computing the dimension of the spaceS ₃ ¹ () of these splines. We consider subspaces ofS ₃ ¹ () satisfying certain boundary conditions. Some applications are given where piecewise cubicC ¹-functions are used to solve interpolation problems and least squares approximation problems.     

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.     

Exponential box splines

In a recent paper by Nira Dyn and the author, univariate cardinal exponential B-splines are shown to have a representation similar to the wellknown box spline representation of the univariate cardinal polynomialB-splines. Motivated by this, we construct, for a set ofn directions inZ ^s and a vector of constants λ ?R ⁿ, an “exponential box spline” which has the same smoothness and support as the polynomial box spline, and is a positive piecewise exponential in its support. We derive recurrence relations for the exponential box splines which are simpler than those for the polynomial case. A relatively simple structure of the space spanned by the translates of an exponential box spline is obtained for λ in a certain open dense set ofR ⁿ—the “simple” λ. In this case, the characterization of the local independence of the translates and related topics, as well as the proofs involved, are quite simple when compared with the polynomial case (corresponding toλ = 0).     

TURBS—Topologically Unrestricted Rational B-Splines

We present a new approach to the construction of piecewise polynomial or rational C ^k -spline surfaces of arbitrary topological structure. The basic idea is to use exclusively parametric smoothness conditions, and to solve the well-known problems at extraordinary points by admitting singular parametrizations. The smoothness of the spline surfaces is guaranteed by specifying a regular smooth reparametrization explicitly. The resulting space of topologically unrestricted rational B-splines (TURBS) is linear and possesses a natural refinement property. Compared with all known methods the construction principle of TURBS is of striking simplicity and the required polynomial bi-degree is essentially decreased from O(k ² ) to d=2k+2 . January 5, 1996. Date revised: September 5, 1996.     

A few remarks on recurrence relations for geometrically continuous piecewise Chebyshevian B-splines

This works complements a recent article (Mazure, J. Comp. Appl. Math. 219(2):457–470, 2008) in which we showed that T. Lyche’s recurrence relations for Chebyshevian B-splines (Lyche, Constr. Approx. 1:155–178, 1985) naturally emerged from blossoms and their properties via de Boor type algorithms. Based on Chebyshevian divided differences, T. Lyche’s approach concerned splines with all sections in the same Chebyshev space and with ordinary connections at the knots. Here, we consider geometrically continuous piecewise Chebyshevian splines, namely, splines with sections in different Chebyshev spaces, and with geometric connections at the knots. In this general framework, we proved in (Mazure, Constr. Approx. 20:603–624, 2004) that existence of B-spline bases could not be separated from existence of blossoms. Actually, the present paper enhances the powerfulness of blossoms in which not only B-splines are inherent, but also their recurrence relations. We compare this fact with the work by G. Mühlbach and Y. Tang (Mühlbach and Tang, Num. Alg. 41:35–78, 2006) who obtained the same recurrence relations via generalised Chebyshevian divided differences, but only under some total positivity assumption on the connexion matrices. We illustrate this comparison with splines with four-dimensional sections. The general situation addressed here also enhances the differences of behaviour between B-splines and the functions of smaller and smaller supports involved in the recurrence relations.     

Multivariate Interpolation with Fundamental Splines of Fractional Order

Fractional B-splines B_σ, σ ≥ 1, are piecewise polynomials of fractional degree that interpolate the classical Schoenberg splines , with respect to the degree. As the Schoenberg splines of order ≥ 3, they in general do not satisfy the interpolation property . However, the application of the interpolation filter —if well-defined—in the frequency domain yields a fundamental spline of fractional order that does satisfy the interpolation property. We extend these result via ridge functions to multivariate fractional B-splines. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Numerical Algorithm for Cubature by Bivariate Splines on Nonuniform Partitions

We have studied the numerical integration 2D based on bivariate C ¹ local polynomial splines with a criss–cross triangulation of nonuniform rectangular partition. We have constructed the cubature formula and proved the convergence properties and error bounds. The paper includes some numerical tests that illustrate the performance of the corresponding algorithm. In the appendix there are explicit expressions of the quadratic polynomial restrictions of the B-splines related to every triangular cell.     

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.     

Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces

Under the appropriate definition of sampling density D_ϕ, a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D_ϕ≥1. This result is similar to Landau's result for the Paley-Wiener space B _1/2 . If the shift invariant space consists of polynomial splines, then we show that D_ϕ<1 is sufficient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B _1/2 .     

Product integration of logarithmic singular integrands based on cubic splines

This paper is concerned with the practical evaluation of the product integral ∫¹_{? 1}f(x)k(x)dx for the case when k(x) = In|x - λ|, λ? (?1, +1) and f is bounded in [?1, +1]. The approximation is a quadrature rule

where the weights {w_n,n,i} are chosen to be exact when f is given by a linear combination of a chosen set of functions {φ_n,j}. In this paper the functions {φ_n,j} are chosen to be cubic B-splines. An error bound for product quadrature rules based on cubic splines is provided. Examples that test the performance of the product quadrature rules for different choices of the function are given. A comparison is made with product quadrature rules based on first kind Chebyshev polynomials.     

On the convexity of C 1 surfaces associated with some quadrilateral finite elements

We analyse the convexity property of two classical finite elements and of the associated piecewise polynomial C ¹ surfaces. The first one is the piecewise cubic quadrilateral of Fraeijs de Veubeke. The second one is a piecewise quadratic rectangle introduced by Sibson and Thomson and generalized by Sablonnière and Zedek. In both cases, we first study the local problem and then we extend our results to the associated C ¹ surfaces. This revised version was published online in June 2006 with corrections to the Cover Date.