Understanding recurrence relations for Chebyshevian B-splines via blossoms

The purpose of this article is to show how naturally recurrence relations for most general Chebyshevian B-splines emerge from blossoms. In particular, this work gives a new insight into previous results by Lyche [A recurrence relation for Chebyshevian B-splines, Constr. Approx. 1 (1985) 155–178], the importance of which it underlines.     

How to build all Chebyshevian spline spaces good for geometric design?

In the present work we determine all Chebyshevian spline spaces good for geometric design. By Chebyshevian spline space we mean a space of splines with sections in different Extended Chebyshev spaces and with connection matrices at the knots. We say that such a spline space is good for design when it possesses blossoms. To justify the terminology, let us recall that, in this general framework, existence of blossoms (defined on a restricted set of tuples) makes it possible to develop all the classical geometric design algorithms for splines. Furthermore, existence of blossoms is equivalent to existence of a B-spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots. We show that Chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Many interesting consequences can be drawn from the latter characterisation: as an example, all Chebsyhevian spline spaces good for design can be built by means of integral recurrence relations.     

On a general new class of quasi Chebyshevian splines

We prove that a general class of splines with sections in different Extended Chebyshev spaces or in different quasi Extended Chebyshev spaces can be viewed as quasi Chebyshevian splines, that is, as splines with all sections in a single convenient quasi Extended Chebyshev space. As a result, we can affirm the presence of blossoms in the corresponding spline spaces, with all the important consequences inherent in blossoms, namely, the possibility of developing all design algorithms for splines, the existence of B-splines bases, along with their optimality.     

A <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Multivariate Clough–Tocher Interpolant

We show that in dimensions four and higher, to insure a smooth interpolant, additional geometric constraints must be imposed on the generalized Clough–Tocher split introduced in Worsey and Farin (Constr. Approx. 3:99–110, [1987]).      

Pairs of Lattice Paths and Positive Trigonometric Sums

Ismail et al. (Constr. Approx. 15:69–81, 1999) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with products of two binomial coefficients.     

Small Support Spline Riesz Wavelets in Low Dimensions

In Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006), a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in Han and Shen (J. Fourier Anal. Appl. 11:615–637, 2005). Motivated by these two papers, we develop in this article a general theory and a construction method to derive small support Riesz wavelets in low dimensions from refinable functions. In particular, we obtain small support spline Riesz wavelets from bivariate and trivariate box splines. Small support Riesz wavelets are desirable for developing efficient algorithms in various applications. For example, the short support Riesz wavelets from Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006) were used in a surface fitting algorithm of Johnson et al. (J. Approx. Theory 159:197–223, 2009), and the Riesz wavelet basis from the Loop scheme was used in a very efficient geometric mesh compression algorithm in Khodakovsky et al. (Proceedings of SIGGRAPH, 2000).     

Statistical Encounters with Complex B-Splines

Complex B-splines as introduced in Forster et al. (Appl. Comput. Harmon. Anal. 20:281–282, 2006) are an extension of Schoenberg’s cardinal splines to include complex orders. We exhibit relationships between these complex B-splines and the complex analogues of the classical difference and divided difference operators and prove a generalization of the Hermite–Genocchi formula. This generalized Hermite–Genocchi formula then gives rise to a more general class of complex B-splines that allows for some interesting stochastic interpretations.      

GB-splines of arbitrary order

Explicit formulae and recurrence relations for the calculation of generalized B-splines (GB-splines) of arbitrary order are given. We derive main properties of GB-splines and their series, i.e. partition of unity, shape-preserving properties, invariance with respect to affine transformations, etc. It is shown that such splines have the variation diminishing property and are Chebyshevian splines.     

On Constrained Nonlinear Hermite Subdivision

We determine shape-preserving regions and we describe a general setting to generate shape-preserving families for the 2-points Hermite subdivision scheme introduced by Merrien (Numer. Algorithms 2:187–200, [1992]). This general construction includes the shape-preserving families presented in Merrien and Sablonníere (Constr. Approx. 19:279–298, [2003]) and Pelosi and Sablonníere (C ¹ GP Hermite Interpolants Generated by a Subdivision Scheme, Prépublication IRMAR 06–23, Rennes, [2006]). New special families are presented as particular examples. Nonstationary and nonuniform versions of such schemes, which produce smoother limits, are discussed.      

Approximation by Discrete GB-Splines

This paper addresses the definition and the study of discrete generalized splines. Discrete generalized splines are continuous piecewise defined functions which meet some smoothness conditions for the first and second divided differences at the knots. They provide a generalization both of smooth generalized splines and of the classical discrete cubic splines. Completely general configurations for steps in divided differences are considered. Direct algorithms are proposed for constructing discrete generalized splines and discrete generalized B-splines (discrete GB-splines for short). Explicit formulae and recurrence relations are obtained for discrete GB-splines. Properties of discrete GB-splines and their series are studied. It is shown that discrete GB-splines form weak Chebyshev systems and that series of discrete GB-splines have a variation diminishing property.     

Normal Multi-scale Transforms for Curves

Extending upon Daubechies et al. (Constr. Approx. 20:399–463, 2004) and Runborg (Multiscale Methods in Science and Engineering, pp. 205–224, 2005), we provide the theoretical analysis of normal multi-scale transforms for curves with general linear predictor S, and a more flexible choice of normal directions. The main parameters influencing the asymptotic properties (convergence, decay estimates for detail coefficients, smoothness of normal re-parametrization) of this transform are the smoothness of the curve, the smoothness of S, and its order of exact polynomial reproduction. Our results give another indication why approximating S may not be the first choice in compression applications of normal multi-scale transforms.     

ECT-B-splines defined by generalized divided differences

ECT-spline curves are generated from different local ECT-systems via connection matrices. If they are nonsingular, lower triangular and totally positive there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized either to form a nonnegative partition of unity or to have integral one. In this paper such ECT-B-splines are defined by generalized divided differences. 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 [Mühlbach and Tang, Calculation of ECT-B-splines and of ECT-spline curves recursively, in preparation].There is an ECT-spline space naturally adjoint to every ECT-spline space. We also construct B-splines via generalized divided differences for this space and study relations between the two adjoint spaces.     

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs

Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair. This research is supported by grants from CNPq and FAPESP.     

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.     

On dimension elevation in Quasi Extended Chebyshev spaces

Via blossoms we analyse the dimension elevation process from to , where is spanned over [0, 1] by 1, x,..., x ^n-2, x ^p , (1 − x) ^q , p, q being any convenient real numbers. Such spaces are not Extended Chebyshev spaces but Quasi Extended Chebyshev spaces. They were recently introduced in CAGD for shape preservation purposes (Costantini in Math Comp 46:203–214; 1986, Costantini in Advanced Course on FAIRSHAPE, pp. 87–114 in 1996; Costantini in Curves and Surfaces with Applications in CAGD, pp. 85–94, 1997). Our results give a new insight into the special case p = q for which dimension elevation had already been considered, first when p = q was supposed to be an integer (Goodman and Mazure in J Approx Theory 109:48–81, 2001), then without the latter requirement (Costantini et al. in Numer Math 101:333–354, 2005). The question of dimension elevation in more general Quasi Extended Chebyshev spaces is also addressed.     

Moments and Fourier transforms of B-splines

The problem of determining the moments and the Fourier transforms of B-splines with arbitrary knots is considered. There exists a simple connection between the moments of such splines and the so-called extended Stirling numbers of the second kind which are defined in section 2. Some recurrence relations for the moments of B-splines with arbitrary knots are given in section 3. In the case of equidistant knots we have also further recurrences. For the forward, central and perfect B-splines the explicit formulas for the moments are given in section 3. The Fourier transforms of B-splines is treated in section 4. The final section is devoted to so-called Stieltjes series connected with the nonnegative weight function w(x) and such that $\text{∫}\text{a}\text{b}\text{w(x) dx}\text{> 0}$ in some closed interval [a, b]. It is proved that such series for the particular values of the independent variable may be expressed by the finite sums which contain the nodes and coefficients of the optimal (in the Davies sense) quadrature formulas.     

Basis of splines associated with some singular differential operators

Functions being piecewise in Ker (D ^k DpD) are a special case of Chebyshev splines having one nontrivial weight and also a special case of singular splines. An algorithm is designed which enables calculating with related B-splines and their derivatives. Ifp(t) is approximated by a piecewise constant, an interesting recurrence for calculating with polynomial B-splines is obtained.