## Bernstein-type operators on a triangle with one curved side.(English)Zbl 1258.41009

The authors construct Bernstein-type operators, and their product and Boolean sum, for a triangle with one curved side. Their interpolation properties and the order of accuracy are studied. Moreover, using the modulus of continuity and Peano’s theorem, respectively, the remainders of the corresponding approximation formulas are also studied. Finally, some numerical examples are given.

### MSC:

 41A36 Approximation by positive operators 41A35 Approximation by operators (in particular, by integral operators) 41A25 Rate of convergence, degree of approximation 41A80 Remainders in approximation formulas
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### References:

 [1] Barnhill R.E.: Surfaces in computer aided geometric design: survey with new results. Comput. Aided Geom. Design 2, 1–17 (1985) · Zbl 0597.65001 · doi:10.1016/0167-8396(85)90002-0 [2] Barnhill R.E., Birkhoff G., Gordon W.J.: Smooth interpolation in triangle. J. Approx. Theory 8, 114–128 (1983) · Zbl 0271.41002 · doi:10.1016/0021-9045(73)90020-8 [3] Barnhill R.E., Gregory I.A.: Polynomial interpolation to boundary data on triangles. Math. Comp., 29(131), 726–735 (1975) · Zbl 0313.65098 · doi:10.1090/S0025-5718-1975-0375735-3 [4] R. E. Barnhill and L. Mansfield, 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. [5] Barnhill R.E., Mansfield L.: Error bounds for smooth interpolation in triangles. J. Approx. Theory 11, 306–318 (1974) · Zbl 0286.41001 · doi:10.1016/0021-9045(74)90002-1 [6] Bernadou M.: C1-curved finite elements with numerical integration for thin plate and thin shell problems, Part 1: construction and interpolation properties of curved C1 finite elements. Comput. Methods Appl. Mech. Engrg. 102, 255–289 (1993) · Zbl 0767.73066 · doi:10.1016/0045-7825(93)90111-A [7] Bernadou M.: C1-curved finite elements with numerical integration for thin plate and thin shell problems, Part 2 : approximation of thin plate and thin shell problems. Comput. Methods Appl. Mech. Engrg. 102, 389–421 (1993) · Zbl 0767.73067 · doi:10.1016/0045-7825(93)90056-4 [8] Blaga P., Coman G.: Bernstein-type operators on triangle. Rev. Anal. Numer. Theor. Approx., 37(1), 9–21 (2009) · Zbl 1212.41066 [9] Blaga P., Cătinaş T., Coman G.: Bernstein-type operators on tetrahedrons. Stud. Univ. Babeş-Bolyai, Mathematica, 54(4), 3–19 (2009) · Zbl 1224.41067 [10] K. Böhmer and G. Coman, Blending interpolation schemes on triangle with error bounds, Lecture Notes in Mathematics, 571, Springer Verlag, Berlin, Heidelberg, New York, 1977, 14–37. · Zbl 0353.41002 [11] Cătinaş T., Coman G.: Some interpolation operators on a simplex domain. Stud. Univ. Babeş–Bolyai Math., 52(3), 25–34 (2007) · Zbl 1164.41300 [12] P. G. Ciarlet, The finite element method for elliptic problems, Noth-Holland, 1978, Reprinted by SIAM, 2002. · Zbl 0383.65058 [13] P. G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Volume II, P.G. Ciarlet and J.L. Lions (eds), North-Holland, 1991, 17-351. · Zbl 0875.65086 [14] Coman G., Blaga P.: Interpolation operators with applications (1). Sci. Math. Jpn., 68(3), 383–416 (2008) · Zbl 1171.41002 [15] Coman G., Blaga P.: Interpolation operators with applications (2). Sci. Math. Jpn., 69(1), 111–152 (2009) · Zbl 1172.41001 [16] Coman G.Cătinaş T.: Interpolation operators on a triangle with one curved side. BIT Numerical Mathematics. 50(2), 243–267 (2010) · Zbl 1200.41002 · doi:10.1007/s10543-010-0256-6 [17] Coman G., Cătinaş T.: Interpolation operators on a tetrahedron with three curved edges. Calcolo, 47(2), 113–128 (2010) · Zbl 1194.41005 · doi:10.1007/s10092-009-0016-7 [18] F. J. Delvos and W. Schempp, Boolean methods in interpolation and approximation, Longman Scientific and Technical, 1989. · Zbl 0698.41032 [19] W. J. Gordon, Distributive lattices and approximation of multivariate functions, Proc. Symp. Approximation with Special Emphasis on Spline Functions (Madison, Wisc.), (Ed. I.J. Schoenberg), 1969, 223–277. · Zbl 0269.41029 [20] Gordon W.J.: Blending-function methods of bivariate and multivariate interpolation and approximation. SIAM J. Numer. Anal., 8, 158–177 (1971) · Zbl 0237.41008 · doi:10.1137/0708019 [21] Nielson G.M., Thomas D.H., Wixom J.A.: Interpolation in triangles. Bull. Austral. Math. Soc., 20(1), 115–130 (1979) · Zbl 0397.65007 · doi:10.1017/S0004972700009138 [22] Renka R.J., Cline A.K.: A triangle-based C1 interpolation method. Rocky Mountain J. Math., 14(1), 223–237 (1984) · Zbl 0568.65006 · doi:10.1216/RMJ-1984-14-1-223 [23] A. Sard, Linear Approximation, American Mathematical Society, Providence, Rhode Island, 1963. · Zbl 0115.05403 [24] L. L. Schumaker, Fitting surfaces to scattered data, Approximation Theory II (G.G. Lorentz, C. K. Chui, L. L. Schumaker, eds.), Academic Press, 1976, 203–268. [25] Stancu D.D.: A method for obtaining polynomials of Bernstein type of two variables. Amer. Math. Monthly 70, 260–264 (1963) · Zbl 0111.26604 · doi:10.2307/2313121 [26] D. D. Stancu, Approximation of bivariate functions by means of some Bernstein-type operators, Multivariate approximation (Sympos., Univ. Durham, Durham, 1977), Academic Press, London-New York, 1978, 189–208.
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