Abstract
We introduce an extension to the two-dimensional simplex of the univariate two-point expansion formula for sufficiently smooth real functions introduced in [13]; it is a polynomial expansion with algebraic degree of exactness. This expansion is applied to obtain a new class of embedded boundary-type cubature formulae on the simplex.
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Abramowitz, M., Stegun, I.A. (eds.): Handbook of mathematical functions with formulas, graphs, and mathematical tables. Washington, D.C.: National Bureau of Standards 1972
Agarwal, R.P., Wong, P.J.Y.: Error inequalities in polynomial interpolation and their applications. Dordrecht: Kluwer 1993
Barnhill, R.E., Birkhoff, G., Gordon, W.J.: Smooth interpolation in triangles. J. Approx. Theory 8, 114–128 (1973)
Barnhill, R.E., Gregory, J.A.: Polynomial interpolation to boundary data on triangles. Math. Comp. 29, 726–735 (1975)
Boas, Jr., R.P.: Representation of functions by Lidstone series. DukeMath. J. 10, 239–245(1943)
Buckholtz, J.D., Shaw, J.K.: On functions expandable in Lidstone series. J. Math. Anal. Appl. 47, 626–632 (1974)
Caira, R., Dell’Accio, F.: Shepard-Bernoulli operators. Math. Comp. 76, 299–321 (2007)
Costabile, F.A., Dell’Accio, F.: Expansions over a simplex of real functions by means of Bernoulli polynomials. Numer. Algorithms 28, 63–86 (2001)
Costabile, F.A., Dell’Accio, F.: Lidstone approximation on the triangle. Appl. Numer. Math. 52, 339–361 (2005)
Costabile, F.A., Dell’Accio, F.: Polynomial approximation of CM function by means of boundary values and applications: a survey. J. Comput. App. Math. 210, 116–135 (2007)
Costabile, F.A., Dell’Accio, F.: New embedded boundary-type quadrature formulas for the simplex. Numer. Algorithms 45, 253–267 (2007)
Costabile, F.A., Dell’Accio, F., Guzzardi, L.: New bivariate polynomial expansion with only boundary data. (Pubblicazioni LAN Unical 15) Arcavacata di Rende: Laboratorio di Analisi Numerica, Università della Calabria 2006
Costabile, F.A., Dell’Accio, F., Luceri, R.: Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values. J. Comput. Appl. Math. 176, 77–90 (2005)
Kergin, P.: A natural interpolation of C K functions. J.Approx. Theory 29, 278–293 (1980)
He, T.-X.: Dimensionality reducing expansion of multivariate integration. Boston: Birkhäuser Boston 2001, pp. 225
Widder, D.V.: Completely convex functions and Lidstone series. Trans. Amer. Math. Soc. 51, 387–398 (1942)
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Costabile, F.A., Dell’Accio, F. & Guzzardi, L. New bivariate polynomial expansion with boundary data on the simplex. Calcolo 45, 177–192 (2008). https://doi.org/10.1007/s10092-008-0149-0
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DOI: https://doi.org/10.1007/s10092-008-0149-0