Sara Remogna

Bivariate C2 cubic spline quasi-interpolants on uniform Powell---Sabin triangulations of a rectangular domain

In this paper we construct discrete quasi-interpolants based on C2 cubic multi-box splines on uniform Powell–Sabin triangulations of a rectangular domain. The main problem consists in finding the coefficient functionals associated with boundary multi-box splines (i.e. multi-box splines whose supports overlap with the domain) involving data points inside or on the boundary of the domain and giving the optimal approximation order. They are obtained either by minimizing an upper bound for the infinity norm of the operator w.r.t. a finite number of free parameters, or by inducing the superconvergence of the gradient of the quasi-interpolant at some specific points of the domain. Finally, we give norm and error estimates and we provide some numerical examples illustrating the approximation properties of the proposed operators.

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

The aim of this paper is to investigate, in a bounded domain of R^3, two blending sums of univariate and bivariate C^1 quadratic spline quasi-interpolants. The main problem consists in constructing the coefficient functionals associated with boundary generators, i.e. generators with supports not entirely inside the domain. In their definition, these functionals involve data points lying inside or on the boundary of the domain. Moreover, the weights of these functionals must be chosen so that the quasi-interpolants have the best approximation order and a reasonable infinite norm. We give their explicit constructions, infinite norms and error estimates. In order to illustrate the approximation properties of the proposed quasi-interpolants, some numerical examples are presented and compared with those obtained by some other trivariate quasi-interpolants given recently in the literature.

B-spline bases for unequally smooth quadratic spline spaces on non-uniform criss-cross triangulations

In this paper, we investigate bivariate quadratic spline spaces on non-uniform criss-cross triangulations of a bounded domain with unequal smoothness across inner grid lines. We provide the dimension of the above spaces and we construct their local bases. Moreover, we propose a computational procedure to get such bases. Finally we introduce spline spaces with unequal smoothness also across oblique mesh segments.

Curve network interpolation by C1 quadratic B-spline surfaces

In this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate C 1 quadratic splines on criss-cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods. Interpolation of a B-spline curve network by a surface based on C 1 quadratic B-splines on criss-cross triangulations.Proof of the existence and uniqueness of the surface and constructive algorithm for its generation.Numerical and graphical results and comparisons with other spline methods.

Near-best C 2 quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains

In this paper, we present new quasi-interpolating spline schemes defined on three-dimensional bounded domains, based on trivariate

Differentiation Based on Optimal Local Spline Quasi‐Interpolants with Applications

C^2

Quasi-interpolation by C1 quartic splines on type-1 triangulations

C2 quartic box splines on type-6 tetrahedral partitions and with approximation order four. Such methods can be used for the reconstruction of gridded volume data. More precisely, we propose near-best quasi-interpolants, i.e. with coefficient functionals obtained by imposing the exactness of the quasi-interpolants on the space of polynomials of total degree three and minimizing an upper bound for their infinity norm. In case of bounded domains the main problem consists in the construction of the coefficient functionals associated with boundary generators (i.e. generators with supports not completely inside the domain), so that the functionals involve data points inside or on the boundary of the domain. We give norm and error estimates and we present some numerical tests, illustrating the approximation properties of the proposed quasi-interpolants, and comparisons with other known spline methods. Some applications with real world volume data are also provided.

On the solution of Fredholm integral equations based on spline quasi-interpolating projectors

The construction of new quasi-interpolants (QIs) having optimal approximation order and small infinity norm and based on a trivariate C 2 quartic box spline is addressed in this paper. These quasi-interpolants, called near-best QIs, are obtained in order to be exact on the space of cubic polynomials and to minimize an upper bound of their infinity norm which depends on a finite number of free parameters in a tetrahedral sequence defining the coefficients of the QIs. We show that this problem has always a unique solution, which is explicitly given. We also prove that the sequence of the resulting near-best quasi-interpolants converges in the infinity norm to the Schoenberg operator.