Francesco A. Costabile

A determinantal approach to Appell polynomials

A new definition by means of a determinantal form for Appell (1880) [1] polynomials is given. General properties, some of them new, are proved by using elementary linear algebra tools. Finally classic and non-classic examples are considered and the coefficients, calculated by an ad hoc Mathematica code, for particular sequences of Appell polynomials are given.

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.

Asymptotic expansion and extrapolation for Bernstein polynomials with applications

Given a real functionf εC2k[0,1],k ≥ 1 and the corresponding Bernstein polynomials {Bn(f)}n we derive an asymptotic expansion formula forBn(f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanBn(f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique.

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operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished).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.

A class of collocation methods for numerical integration of initial value problems

For the numerical solution of initial value problems a general procedure to determine global integration methods is derived and studied. They are collocation methods which can be easily implemented and provide a high order accuracy. They further provide globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. Numerical experiments provide favorable comparisons with other existing methods.

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.

An algebraic approach to Lidstone polynomials

Abstract A new definition of Lidstone polynomials [G.L. Lidstone, Note on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types, Proc. Edinb. Math. Soc. 2 (2) (1929) 16–19] is proposed; this is a Hessemberg determinantal form. The algebraic approach provides an elementary proof of the main recursive properties of Lidstone polynomials.

An algebraic approach to Sheffer polynomial sequences

A matrix approach to Sheffer polynomial sequences is proposed; in particular, two different determinantal forms of Sheffer sequences are given, the one as the function of a polynomial sequence of binomial type and the other as the function of the canonical base xi. The equivalence with the classical definitions of Sheffer and Roman and Rota is proven. Then, elementary matrix algebra tools are employed to reveal the known and unknown properties of Sheffer polynomials. Finally, classical and non-classical examples are also considered.

Collocation for High-Order Differential Equations with Lidstone Boundary Conditions

A class of methods for the numerical solution of high-order differential equations with Lidstone and complementary Lidstone boundary conditions are presented. It is a collocation method which provides globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients is generated by a recurrence formula. Numerical experiments support theoretical results.