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In this paper, we use the finite element method to construct a new normalized basis of a univariate quadratic $C^1$ spline space. We give a new representation of Hermite interpolant of any piecewise polynomial of class at least $C^1$ in terms of its polar form. We use this representation for constructing several superconvergent and super-superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.

In this paper, we use the finite element method to construct a new normalized basis of a univariate C 2 cubic spline space endowed with a specific subdivision of a real interval. Based on the polar forms, we introduce a new representation of the Hermite interpolant of any C 2 piecewise polynomial defined over this subdivision and we construct several superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.

The paper is concerned with a multipoints iterative method for approximating a simple zero z of a real valued function. This method is essentially based on the idea to extend secant method ...

This work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions (splines), which uses the blossoming approach. Some existing results in the literature are reformulated, such as the smoothness conditions between polynomials of a spline, by using the affinity property of the blossom. Some definitions of sub-splines are proposed which can be very useful in the study and the construction of splines such as macro-elements or quasi-interpolants. As an application of the proposed results, a C 1 quartic spline quasi-interpolant with optimal approximation order is defined without using any mask for smoothness or B-spline basis. Numerical results are presented and compared with other methods given in the literature.

A new approach to construct univariate spline quasi-interpolants on arbitrary partitions of bounded intervals is developed. In the first part of this paper, we give some results about the symmetric functions of the difference of two finite sets. These results are used, in the second part of this work, to construct explicitly different types of quasi-interpolants. We revise the definition of a uniformly bounded quasi-interpolant and we propose some results on this subject. Some numerical examples are given to illustrate our theoretical results.

In this paper, we use C 1 cubic B-splines to construct the Hermite interpolant of any polynomial in terms of their blossom. Consequently, a simple method is presented to get superconvergence phenomenon of cubic spline quasi-interpolants at the knots of a uniform partition. Thanks to this phenomenon, the cubic spline quasi-interpolant provides an interesting approximation very accurate at the superconvergence points. Numerical results are given to illustrate the theoretical ones.

In this work we describe an approximating scheme based on simplex splines on a tetrahedral partition using volumetric data. We use the trivariate simplex B-spline representation of polynomials or piecewise polynomials in terms of their polar forms to construct several differential or discrete quasi interpolants which have an optimal approximation order.

In this paper, we describe an iterative method for approximating asimple zero $z$ of a real defined function. This method is aessentially based on the idea to extend Newton's method to be theinverse quadratic interpolation. We prove that for a sufficientlysmooth function $f$ in a neighborhood of $z$ the order of theconvergence is quartic. Using Mathematica with its high precisioncompatibility, we present some numerical examples to confirm thetheoretical results and to compare our method with the others givenin the literature.

In this paper, we study a new method for function approximation from the given integral values over successive subintervals by using cubic B-splines. The method does not need any additional end conditions and it is easy to be implemented without solving any system of linear equations. The method is able to approximate the original function and its first and second-order derivatives over the global interval successfully. The approximation errors are well studied. Numerical results illustrate that our method is very effective.

In this paper we use Normalized trivariate Worsey-Piper B-splines recently constructed by Sbibih et?al. (2012) and the method proposed in Sbibih et?al. (2013) to give a new representation of Worsey-Piper Hermite interpolant of any piecewise polynomial of class at least C 1 over the Worsey-Piper split in terms of its polar forms. Using this representation we construct several superconvergent discrete quasi-interpolants. The construction that we present in this work is a generalization of the one presented in Sbibih et?al. (2012) with other properties.

In this paper, we first recall some results concerning the construction and the properties of quadratic B-splines over a refinement Δ of a quadrangulation ◊ of a planar domain introduced recently by Lamnii et al. Then we introduce the B-spline representation of Hermite interpolant, in the special space S 2 1 , 0 ( Δ ) , of any polynomial or any piecewise polynomial over refined quadrangulation Δ of ◊ . After that, we use this B-representation for constructing several superconvergent discrete quasi-interpolants. The new results that we present in this paper are an improvement and a generalization of those developed in the above cited paper.

In this paper we use Normalized Powell–Sabin B-splines constructed by Dierckx [6] to introduce a new B-spline representation of Hermite Powell–Sabin interpolant of any polynomial or any piecewise polynomial over Powell–Sabin partitions of class at least C 1 in terms of their polar forms. We use this representation for constructing several superconvergent discrete quasi-interpolants. The result that we present in this paper is a generalization of the one presented in [20] with other properties.

In this paper, we present a new technique to get superconvergence phenomenon of cubic spline quasi-interpolants at the knots of an uniform partition. This method gives rise to good approximation not only at these knots but also on the whole domain of definition. Moreover, We give an application to numerical integration. Numerical results are given to illustrate the theoretical ones.

In this paper we introduce a B-spline representation of the cubic Hermite Powell-Sabin interpolant of any polynomial or any piecewise polynomial, over Powell-Sabin partitions of class at least $$C^1$$ , in terms of their polar forms. We use this B-spline representation for constructing several superconvergent discrete cubic spline quasi-interpolants which approximate a function $$f$$ better than the superconvergent quadratic ones developed in one of our recent published papers. The new results presented in this work are an improvement and a generalization of those studied recently in the literature. We also illustrate by numerical examples that global errors and cubature rules based on these cubic Powell-Sabin spline quasi-interpolants are positively affected by the superconvergence phenomenon.

Let $${\fancyscript{S}}(\phi _m)$$ be the space generated by a finite set $$\phi _m$$ of continuous functions defined on a domain $$\varOmega $$ in $$\mathbb {R}^s$$ . We suppose that this space contains the space of polynomials of degree at most $$m$$ . By using the blossoming approach, we show how to construct multivariate quasi-interpolants which have important properties such as high order of regularity and polynomial reproduction. The quasi-interpolation coefficients are polynomials, obtained as a blossom of a specific polynomial. We will show that some results existing in the literature can be obtained as particular cases to our method.

By blossoming Marsden’s identity, we investigate local quasi-interpolation schemes for C 2 -continuous quintic Powell–Sabin splines represented with a normalized B-spline basis. As applications, various families of discrete and differential quasi-interpolants reproducing quintic polynomials are presented.

In this paper, we construct a quadratic composite finite element of class C1 and quartic composite finite element of class C2 on a new triangulation τ10 which is obtained by splitting each triangle of a given triangulation τ into ten smaller subtriangles. These new elements can be used for constructing spline spaces with local basis that can be applied for solving some Hermite interpolation problems with optimal approximation order.

In this work, we give an algorithm for constructing a normalized B-spline basis over a Worsey-Piper split of a bounded domain of ℝ3. These B-splines are all positive, have local support and form a partition of unity. Therefore, they can be used for constructing local approximants and for many other applications in CAGD. We also introduce the Worsey-Piper B-spline representation of C 1 quadratic polynomials or splines in terms of their polar forms. Then, we use this B-representation for constructing several quasi-interpolants which have an optimal approximation order.

In this paper we present a method to obtain a trivariate spline constructed over the Worsey-Piper split corresponding to a tetrahedron. Such spline approximates a set of Lagrangian scattered data by minimizing an ''energy functional'' which also controls the smoothness of the spline. We give a convergence result and we show some graphical examples.