In this work, we are interested in the resolution of the Laplace equation -Delta u=f with Dirichlet boundary condition in a closed surface S in R-2, which is - topologically - equivalent to the unit disc D={(x, y) |x(2)+y(2) <= 1}. It is known that for a function u represented in polar coordinates on D, certain boundary conditions must be satisfied by u so that the surface S is of class C-0. More precisely, we construct an approximant of class C-0 on D as a tensor product of two quasi-interpolants, one based on UAT-splines and the other based on classical B-splines. Some numerical results are given to validate the work.

This paper focuses on the approximation of solutions of second kind Fredholm integral equations using non-uniform spline quasi-interpolation. Our aim is to determine the most effective non-uniform partition that provides an optimal numerical solution to the integral equation. To achieve this, we introduce a solution approach based on genetic algorithms, using right approximation of the integral equation’s kernel. We present some numerical examples to show the method’s performance.

The aim of this paper is to present a family of polynomial quasi-interpolants in Bernstein basis. More precisely, we will combine the strong features of the polar forms and the symmetric polynomials to derive the coefficients of the quasi-interpolant in the Bernstein basis representation. We also derive a collection of spline quasi-interpolants that reproduce polynomial functions up to degree 2. Numerical examples support the theoretical results and show that the proposed scheme is simple and effective.

In this paper, we introduce and study new h-Bernstein basis functions over a triangular domain. In particular, after defining the h-Bernstein polynomial functions of degree n, we prove their algebraic and geometric properties, such as partition of unity and degree elevation and we show that they form a basis for the space of polynomials of total degree less than or equal to n on a triangle. Then, we propose the h-de Casteljau algorithm and we prove the Marsden identity.

In this paper, a construction of Marsden’s identity for UAH B-splines (i.e. Uniform Algebraic Hyperbolic B-splines) is developed and a clear proof is given. With the help of this identity, quasi-interpolant schemes that produce the space of algebraic hyperbolic functions are derived. Efficient quadrature rules, based on integrating some of these quasi-interpolant schemes, are constructed and studied. Numerical results that illustrate the effectiveness of these rules are presented.

In this paper, we use a cubic spline collocation method to solve a two dimensional convection–diffusion equation. More precisely, we approximate first and second order partial derivatives by those of cubic spline quasi-interpolants to produce a system of first order ordinary differential equations. The resulting system can be solved using MATLAB’s ode solver. Error estimates of quasi-interpolants which are used are given with full discussion. Furthermore, numerical examples are presented to show the validity of our methods.

The aim of this paper is to present an Hermite interpolation problem with B-splines of high degree of smoothness. More precisely, we use polar forms to find the B-spline control points for Hermite interpolation. The resulting formula is used to give Hermite basis functions. In particular, Quadratic C 1 and cubic C 2 interpolations with sharp parameters are analyzed.

In this paper we present a polynomial basis based on two-point osculatory interpolation. By exploring some interesting properties of this basis, we derive the smoothness conditions. These conditions can be used for the construction of smooth splines with a low polynomial degree in terms of data points. As an application we give an efficient method for constructing composite splines with shape parameters.

In this paper, we show how to construct a normalized B-spline basis for a special C1 continuous splines of degree 2, defined on Sibson–Thomson refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The dilatation equation can be found by applying the dyadic subdivision scheme directly to the Sibson–Thomson spline basis functions. As an application, a quasi-interpolation method, based on this Sibson–Thomson B-spline representation, is described which can be used for the efficient visualization of gridded surface data.

The purpose of this article is the construction of a normalized basis for a quadratic condensed Powell–Sabin-12 macro-element space introduced by Alfeld et al. (2010). The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction of this basis is adopted from Dierckx (1997) and Speleers (2010a), and is based on the determination of a set of triangles that must contain a specific set of points. The proposed basis can only be constructed on triangulations with a maximal angle less than π2.

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 using the polarization identity, we propose a family of quasi-interpolants based on bivariate \({\fancyscript{C}}^1\) cubic super splines defined on triangulations with a Powell–Sabin refinement. Their spline coefficients only depend on a set of local function values. The quasi-interpolants reproduce cubic polynomials and have an optimal approximation order.

In this paper, we describe the construction of a suitable normalized B-spline representation for bivariate C1 cubic super splines defined on triangulations with a Powell–Sabin refinement. The basis functions have local supports, they form a convex partition of unity, and every spline is locally controllable by means of control triangles. As application, discrete and differential quasi-interpolants of optimal approximation order are developed and numerical tests for illustrating theoretical results are presented.

By blossoming Marsden’s identity, we investigate local quasi-interpolation schemes forC2-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 describe the construction of a suitable normalized B-spline representation for special multivariate quadratic spline space S"2^1^,^0(@D) over a refined quadrangulation. We then develop as an application, a general theory of quasi-interpolants based on this representation.