A mixed Lagrange-Bernoulli tensor product expansion on the rectangle with applications

Abstract A mixed Lagrange–Bernoulli tensor product polynomial operator for bivariate functions of class ( m , n ) with only boundary data on the rectangle is presented. Properties of interpolation and error bounds are studied. Application to functions approximation, solution of Poisson equations with Dirichelet’s conditions and numerical cubature are proposed. Numerical results are also given.

[1]  Francesco Dell'Accio,et al.  Complete Hermite-Birkhoff interpolation on scattered data by combined Shepard operators , 2016, J. Comput. Appl. Math..

[2]  Charles K. Chui,et al.  Multivariate vertex splines and finite elements , 1990 .

[3]  Francesco Dell'Accio,et al.  Complementary Lidstone interpolation on scattered data sets , 2012, Numerical Algorithms.

[4]  F. Costabile,et al.  Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values , 2005 .

[5]  F. A. Costabile,et al.  Expansion Over a Rectangle of Real Functions in Bernoulli Polynomials and Applications , 2001 .

[6]  Francesco Dell'Accio,et al.  On the bivariate Shepard-Lidstone operators , 2012, J. Comput. Appl. Math..

[7]  A. C. Ahlin,et al.  A bivariate generalization of Hermite’s interpolation formula , 1964 .

[8]  Francesco Dell'Accio,et al.  Enhancing the approximation order of local Shepard operators by Hermite polynomials , 2012, Comput. Math. Appl..

[9]  Francesco A. Costabile,et al.  The Appell interpolation problem , 2011, J. Comput. Appl. Math..

[10]  G. J. Lidstone.  Notes on the Extension of Aitken's Theorem (for Polynomial Interpolation) to the Everett Types. , 1930 .

[11]  K. Hormann,et al.  On the approximation order of triangular Shepard interpolation , 2015 .

[12]  Francesco Dell'Accio,et al.  Polynomial approximation of CM functions by means of boundary values and applications: A survey , 2007 .

[13]  Francesco Dell'Accio,et al.  A new approach to Bernoulli polynomials , 2006 .

[14]  R. Varga,et al.  Piecewise Hermite interpolation in one and two variables with applications to partial differential equations , 1968 .

[15]  A. Sharma,et al.  Modified Abel Expansion and a Subclass of Completely Convex Functions , 1972 .