Bernstein-type operators on triangles

Authors

  • Petru Blaga Babeş-Bolyai University, Cluj-Napoca, Romania
  • Gheorghe Coman Babeş-Bolyai University, Cluj-Napoca, Romania

DOI:

https://doi.org/10.33993/jnaat381-898

Keywords:

Bernstein operator, product operator, Boolean sum operator, modulus of continuity, error evaluation
Abstract views: 230

Abstract

The aim of the paper is to construct some univariate Bernstein-type operators on triangle, their product and Boolean sum, which interpolate a given function on the edges respectively at the vertices of triangle. Using the modulus of continuity and the Peano's theorem the remainders of corresponding approximation formulas are studied.

Downloads

Download data is not yet available.

References

Barnhill, R. E., Surfaces in computer aided geometric design: survey with new results, Comput. Aided Geom. Design, 2, pp. 1-17, 1985, https://doi.org/10.1016/0167-8396(85)90002-0 DOI: https://doi.org/10.1016/0167-8396(85)90002-0

Barnhill, R. E., Birkhoff, G. and Gordon,W. J., Smooth interpolation in triangle, J. Approx. Theory, 8, pp. 114-128, 1973, https://doi.org/10.1016/0021-9045(73)90020-8 DOI: https://doi.org/10.1016/0021-9045(73)90020-8

Barnhill, R. E. and Gregory, J. A., Polynomial interpolation to boundary data on triangles, Math. Comp., 29, no. 131, pp. 726-735, 1975, https://doi.org/10.1090/s0025-5718-1975-0375735-3 DOI: https://doi.org/10.1090/S0025-5718-1975-0375735-3

Barnhill, R. E. and Mansfield, L., Error bounds for smooth interpolation in triangles, J. Approx. Theory, 11, pp. 306-318, 1974, https://doi.org/10.1016/0021-9045(74)90002-1 DOI: https://doi.org/10.1016/0021-9045(74)90002-1

Barnhill, R. E. and Mansfield, L., Sard kernel theorems on triangular and rectangular domains with extensions and applications to finite element error, Technical Report 11, Department of Mathematics, Brunel Univ., 1972.

Bernardi, C., Optimal finite-element interpolation one curved domains, SIAM J. Numer. Anal. 26, no. 5, pp. 1212-1240, 1989, https://doi.org/10.1137/0726068 DOI: https://doi.org/10.1137/0726068

Böhmer, K. and Coman, Gh., Blending interpolation schemes on triangle with error bounds, Lecture Notes in Mathematics, 571, Springer Verlag, Berlin, Heidelberg, New York, pp. 14-37, 1977, https://doi.org/10.1007/bfb0086562 DOI: https://doi.org/10.1007/BFb0086562

Bojanov, B. and Xu, Y., On polynomial interpolation of two variables, J. Approx. Theory, 120, no. 2, pp. 267-282, 2003, https://doi.org/10.1016/s0021-9045(02)00023-0 DOI: https://doi.org/10.1016/S0021-9045(02)00023-0

Cătinaş, T. and Coman, Gh., Some interpolation operators on a simplex domain, Stud. Univ. Babeş-Bolyai Math., 52, no. 3, pp. 25-34, 2007.

Coman, G. and Blaga, P., Interpolation operators with applications (1), Sci. Math. Jpn., 68, no. 3, pp. 383-416, 2008.

Coman, G. and Blaga, P., Interpolation operators with applications (2), Sci. Math. Jpn., 69, no. 1, pp. 111-152, 2009.

Coman, Gh., Cătinaş, T., Birou, M., Oprişan, A., Oşan, C., Pop, I., Somogyi, I. and Todea, I., Interpolation Operators, Ed. Casa Cărţii de Ştiinţă, Cluj-Napoca, 2004.

Coman, Gh. and Gânscă, I., Blending approximation with applications in construction, Bul. Şt. Inst. Pol. Cluj-Napoca, 24, pp. 35-40, 1981.

Coman, Gh., Gânscă, I. and Ţâmbulea, L., Surfaces generated by blending interpolation, Stud. Univ. Babeş-Bolyai Math., 38, pp. 39-48, 1993.

Coons, S. A., Surface for computer aided design of space forms, Project MAC, Design Div., Dep. of Mech. Engineering, MIT, 1964.

Hall, C. A., Bicubic interpolation over triangles, J. Math. Mech., 19, pp. 1-11, 1969, https://doi.org/10.1512/iumj.1970.19.19001 DOI: https://doi.org/10.1512/iumj.1970.19.19001

Hulme, B. L., A new bicubic interpolation over right triangle, J. Approx. Theory, 5, pp. 66-73, 1972, https://doi.org/10.1016/0021-9045(72)90029-9 DOI: https://doi.org/10.1016/0021-9045(72)90029-9

Marshall, J. A. and Mitchell A. R., An exact boundary technique for improved accuaracy in the finite element method, J. Inst. Math. Appl., 12, pp. 355-362, 1973, https://doi.org/10.1093/imamat/12.3.355 DOI: https://doi.org/10.1093/imamat/12.3.355

Marshall, J. A. and Mitchell A. R., Blending interpolants in the finite element method, Internat. J. Numer. Methods Engrg., 12, pp. 77-83, 1978, https://doi.org/10.1002/nme.1620120108 DOI: https://doi.org/10.1002/nme.1620120108

Mitchell A. R. and McLeod, R., Curved elements in the finite element method, Conference on Numer. Sol. Diff. Eq. Lecture Notes in Mathematics, 363, Springer Verlag, pp. 89-104, 1974, https://doi.org/10.1007/bfb0069128 DOI: https://doi.org/10.1007/BFb0069128

Nielson, G. M., Thomas, D. H. and Wixom, J. A., Interpolation in triangles, Bull. Austral. Math. Soc., 20, no. 1, pp. 115-130, 1979, https://doi.org/10.1017/s0004972700009138 DOI: https://doi.org/10.1017/S0004972700009138

Renka, R. J. and Cline, A. K., A triangle-based C¹ interpolation method, Rocky Mountain J. Math., 14, no. 1, pp. 223-237, 1984, https://doi.org/10.1216/rmj-1984-14-1-223 DOI: https://doi.org/10.1216/RMJ-1984-14-1-223

Sard, A., Linear Approximation, American Mathematical Society, Providence, Rhode Island, 1963. DOI: https://doi.org/10.1090/surv/009

Schumaker, L. L., Fitting surfaces to scattered data, Approximation Theory II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, eds.), Academic Press, pp. 203-268, 1976. DOI: https://doi.org/10.21236/ADA027870

Sendov, B. and Andreev, A., Approximation and interpolation theory. Handbook of numerical analysis (P. G. Ciarlet and J. L. Lions, eds.) Vol. III, pp. 223-462, North-Holland, Amsterdam, 1994, https://doi.org/10.1016/s1570-8659(05)80017-1 DOI: https://doi.org/10.1016/S1570-8659(05)80017-1

Stancu, D. D., Généralisation de certaines formules d'interpolation pour les fonctions de plusieurs variables; quelques considérations sur la formule d'intégration numérique de Gauss (Romanian), Acad. R. P. Romîne. Bul. Sti. Sect. Sti. Mat. Fiz., 9, pp. 287-313, 1957.

Stancu, D. D., A method for obtaining polynomials of Bernstein type of two variables, Amer. Math. Monthly, 70, pp. 260-264, 1963, https://doi.org/10.1080/00029890.1963.11990079 DOI: https://doi.org/10.1080/00029890.1963.11990079

Stancu, D. D., Evaluation of the remainder term in approximation formulas by Benstein polynomials, Math. Comp., 17, pp. 270-278, 1963, https://doi.org/10.1090/s0025-5718-1963-0179524-6 DOI: https://doi.org/10.1090/S0025-5718-1963-0179524-6

Stancu, D. D., The remainder of certain linear approximation formulas in two variables, SIAM Numer. Anal. Ser. B, 1, pp. 137-163, 1964, https://doi.org/10.1137/0701013 DOI: https://doi.org/10.1137/0701013

Stancu, D. D., Approximation of bivariate functions by means of some Bernstein-type operators, Multivariate approximation (Sympos., Univ. Durham, Durham, 1977), pp. 189-208, Academic Press, London-New York, 1978.

Stancu, D. D., Coman, G., Agratini, O. and Trîmbiţaş, R., Numerical analysis and approximation theory, Vol. I (Romanian), Presa Universitară Clujeană, Cluj-Napoca, 2001.

Steffensen, J. F., Interpolation, 2d ed., Chelsea Publishing Co., New York, 1950.

Zlamal, M., Curved elements in the finite element method, SIAM Numer. Anal., 10, pp. 229-240, 1973, https://doi.org/10.1137/0710022 DOI: https://doi.org/10.1137/0710022

Downloads

Published

2009-02-01

How to Cite

Blaga, P., & Coman, G. (2009). Bernstein-type operators on triangles. Rev. Anal. Numér. Théor. Approx., 38(1), 9–21. https://doi.org/10.33993/jnaat381-898

Issue

Vol. 38 No. 1 (2009)

Section

Articles

License

Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.