Cheney–Sharma type operators on a triangle with two and three curved edges

Alina  Baboş

doi:10.37863/umzh.v72i5.6017

Alina Baboş Dep. Tech. Sci., "Nicolae Balcescu" Land Forces Academy, Sibiu, Romania

DOI: https://doi.org/10.37863/umzh.v72i5.6017

Abstract

UDC 517.5

We construct some Cheney–Sharma type operators de ned on a triangle with two and three curved edges, their product and Boolean sum. We study their interpolation properties and the degree of exactness.

References

Baboş, A. Some interpolation operators on triangle. The 16th Int. Conf. the Knowledge-Based Organization, Appl. Tech. Sci. and Adv. Military Technologies , Sibiu (2010), p. 28–34.

Baboş, A. Some interpolation schemes on a triangle with one curved side. Gen. Math. 21 (2013), no. 1-2, 97–106.

Baboş, A. Interpolation operators on a triangle with two and three edges. Creat. Math. Inform. 22 (2013), no. 2, 135–142.

Barnhill, R. E.; Gregory, J. A. Polynomial interpolation to boundary data on triangles. Math. Comp. 29 (1975), 726–735. https://doi.org/10.1090/s0025-5718-1975-0375735-3 DOI: https://doi.org/10.1090/S0025-5718-1975-0375735-3

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

Barnhill, Robert E.; Mansfield, Lois. Error bounds for smooth interpolation in triangles. J. Approximation Theory https://doi.org/10.1016/0021-9045(74)90002-1 DOI: https://doi.org/10.1016/0021-9045(74)90002-1

Bărbosu, Dan; Zelina, Ioana. About some interpolation formulas over triangles. Rev. Anal. Numér. Théor. Approx. 28 (1999), no. 2, 117–123 (2000). https://ictp.acad.ro/jnaat/journal/article/view/1999-vol28-no2-art2

Bernardi, Christine. Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989), no. 5, 1212–1240. https://doi.org/10.1137/0726068 DOI: https://doi.org/10.1137/0726068

Birkhoff, Garrett. Interpolation to boundary data in triangles. Collection of articles dedicated to Salomon Bochner. J. Math. Anal. Appl. 42 (1973), 474–484. https://doi.org/10.1016/0022-247X(73)90154-6 DOI: https://doi.org/10.1016/0022-247X(73)90154-6

Blaga, Petru; Coman, Gheorghe. Bernstein-type operators on triangles. Rev. Anal. Numér. Théor. Approx. 38 (2009), no. 1, 11–23 (2010).

Blaga, Petru; Cătinaş, Teodora; Coman, Gheorghe. Bernstein-type operators on a triangle with one curved side. Mediterr. J. Math. 9 (2012), no. 4, 833–845. https://doi.org/10.1007/s00009-011-0156-2 DOI: https://doi.org/10.1007/s00009-011-0156-2

Blaga, Petru; Cătinaş, Teodora; Coman, Gheorghe. Bernstein-type operators on a triangle with all curved sides. Appl. Math. Comput. 218 (2011), no. 7, 3072–3082. https://doi.org/10.1016/j.amc.2011.08.027 DOI: https://doi.org/10.1016/j.amc.2011.08.027

Blaga, Petru; Cătinaş, Teodora; Coman, Gheorghe. Bernstein-type operators on tetrahedrons. Stud. Univ. Babeş-Bolyai Math. 54 (2009), no. 4, 3–18. http://www.cs.ubbcluj.ro/~studia-m/2009-4/blaga-final.pdf

Cătinaş, Teodora; Coman, Gheorghe. Some interpolation operators on a simplex domain. Stud. Univ. Babeş-Bolyai Math. 52 (2007), no. 3, 25–34. http://www.cs.ubbcluj.ro/~studia-m/2007-3/catinas.pdf

Coman, Gheorghe; Cătinaş, Teodora. Interpolation operators on a triangle with one curved side. BIT 50 (2010), no. 2, 243–267. https://doi.org/10.1007/s10543-010-0256-6 DOI: https://doi.org/10.1007/s10543-010-0256-6

Cheney, E. W.; Sharma, A. On a generalization of Bernstein polynomials. Riv. Mat. Univ. Parma (2) 5 (1964), 77–84. http://www.rivmat.unipr.it/fulltext/1964-5/1964-5-077.pdf

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

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

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

Mitchell, A. R.; McLeod, R. Curved elements in the finite element method. Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973), pp. 89–104. Lecture Notes in Math., Vol. 363, Springer, Berlin, 1974. https://doi.org/10.1007/bfb0069128 DOI: https://doi.org/10.1007/BFb0069128