F. Dell'Accio

Expansion Over a Rectangle of Real Functions in Bernoulli Polynomials and Applications

In this paper we generalize an expansion in Bernoulli polynomials for real functions possessing a sufficient number of derivatives. Starting from this expansion we obtain useful kernels, which are substantially different from Sards for a wide class of linear functionals that includes the truncation error for cubature formulas.

Shepard–Bernoulli operators

We introduce the Shepard-Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the interpolation of discrete solutions of initial value problems is given.

Complementary Lidstone interpolation on scattered data sets

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691–1707, 2012) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell’Accio, Appl Numer Math 52:339–361, 2005). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77–90, 2005) and drawn on by Agarwal et al. (2009) and Agarwal and Wong (J Comput Appl Math 234:2543–2561, 2010). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided.

Enhancing the approximation order of local Shepard operators by Hermite polynomials

We show how to combine local Shepard operators with Hermite polynomials on the simplex [C. K. Chui, M.-J. Lai, Multivariate vertex splines and finite elements, J. Approx. Theory 60 (1990) 245-343] so as to raise the algebraic precision of the Shepard-Taylor operators [R. Farwig, Rate of convergence of Shepards global interpolation formula, Math. Comp. 46 (1986) 577-590] that use the same data and contemporaneously maintain the interpolation properties at each sample point (derivative data included) and a good accuracy of approximation. Numerical results are provided.

Expansions over a Simplex of Real Functions by Means of Bernoulli Polynomials

In [1] there is an expansion in Bernoulli polynomials for sufficiently smooth real functions in an interval [a,b]⊂R that has useful applications to numerical analysis. An analogous result in a 2-dimensional context is derived in [2] in the case of rectangle. In this note we generalize the above-mentioned one-dimensional expansion to the case of Cm-functions on a 2-dimensional simplex; a method to generalize the expansion on an N-dimensional simplex is also discussed. This new expansion is applied to find new cubature formulas for 2-dimensional simplex.

Complete Hermite-Birkhoff interpolation on scattered data by combined Shepard operators

The problem of Hermite-Birkhoff interpolation on scattered data under certain conditions of completeness is considered by using Shepard basis functions in combination with local interpolating polynomials based on the vertices of triangles. This approach allows the construction of interpolation operators with nontrivial polynomial reproduction and a good accuracy of approximation.

On the constrained mock-Chebyshev least-squares

The algebraic polynomial interpolation on n + 1 uniformly distributed nodes can be affected by the Runge phenomenon, also when the function f to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which produces a polynomial P that interpolates f on a subset of m + 1 of the given nodes whose elements mimic as well as possible the Chebyshev-Lobatto points of order m . In this work we use the simultaneous approximation theory to produce a polynomial P ? of degree r , greater than m , which still interpolates f on the m + 1 mock-Chebyshev nodes minimizing, at the same time, the approximation error in a least-squares sense on the other points of the sampling grid. We give indications on how to select the degree r in order to obtain polynomial approximant good in the uniform norm. Furthermore, we provide a sufficient condition under which the accuracy of the mock-Chebyshev interpolation in the uniform norm is improved. Numerical results are provided.

Relation between grid, channel, and Peano networks in high‐resolution digital elevation models

The topological interconnection between grid, channel, and Peano networks is investigated by extracting grid and channel networks from high-resolution digital elevation models of real drainage basins, and by using a perturbed form of the equation describing how the average junction degree varies with Horton-Strahler order in Peano networks. The perturbed equation is used to fit the data observed over the Hortonian substructures of real networks. The perturbation parameter, denoted as “uniformity factor,” is shown to indicate the degree of topological similarity between Hortonian and Peano networks. The sensitivities of computed uniformity factors and drainage densities to grid cell size and selected threshold for channel initiation are evaluated. While the topological relation between real and Peano networks may not vary significantly with grid cell size, these networks are found to exhibit the same drainage density only for specific grid cell sizes, which may depend on the selected threshold for channel initiation.

Special issue on New Trends in Numerical Analysis: Theory, Methods, Algorithms and Applications (NETNA2015)

Special issue on New Trends in Numerical Analysis : Theory, Methods, Algorithms and Applications (NETNA2015) Preface

The numerical calculation of topological turning points

We introduce the definition of topological turning point of a function ℱ(x, λ): ℝ×ℝ→ℝ, then we propose a numerical method for calculating it. This new definition does not require any regularity for ℱ but its continuity; moreover, topological turning point coincides with turning point when ℱ is sufficiently smooth. The numerical method that we introduce has linear rate of convergence, and it is of secure convergence.