On estimation of the uniform approximation error by interpolating multilinear spline with $l_p$ distances

Laiyi Zhu; Guoyi Yang

doi:10.3842/umzh.v76i8.7532

Laiyi Zhu School of Mathematics, Renmin University of China, Beijing, China
Guoyi Yang School of Mathematics, Renmin University of China, Beijing, China

DOI: https://doi.org/10.3842/umzh.v76i8.7532

Keywords: Multilinear splines; Interpolation; Uniform approximation error; $l_p$ distances; Class of function.

Abstract

UDC 517.5

We consider the estimation of the uniform approximation error on classes of functions by interpolating multilinear spline with $l_p$ distances for $p>3.$

References

R. Anderson, Y. Babenko, T. Leskevych, Simultaneous approximation of a multivariate function and its derivatives by multilinear splines, J. Approx. Theory, 183, 82–97 (2014). DOI: https://doi.org/10.1016/j.jat.2014.04.013

N. P. Korneychuk, Sharp constants in approximation theory, in: Encyclopedia of Mathematics and Its Applications, vol. 38, Cambridge Univ. Press (1991).

V. N. Malozemov, On the deviation of broken lines} (in Russian), Vestn. Leningr. Univ. Math. Mekh. Astron., 7, 150–153 (1966).

V. N. Malozemov, On polygonal interpolation, Math. Notes, 1, № 5, 537–540 (1967). DOI: https://doi.org/10.1007/BF01094071

Sh. E. Mikeladze, Numerical methods of mathematical analysis} (in Russian), Gostekhizdat, Moscow (1953).

M. Sh. Shabozov, On the error of the interpolation by bilinear splines, Ukr. Math. J., 46, № 11, 1719–1726 (1994). DOI: https://doi.org/10.1007/BF01058889

V. F. Storchai, Approximation of continuous functions of two variables by splines in metric $C$, in: Investigations in Contemporary Problems of Summation and Approximation of Functions and Their Applications (in Russian), Dnepropetrovsk State University, Dnepropetrovsk (1972), p. 66–68.

V. F. Storchai, Approximation of continuous functions of two variables by polyhedral functions and splines in the uniform metric, in: Investigations in Contemporary Problems of Summation and Approximation of Functions and Their Applications (in Russian), Dnepropetrovsk State University, Dnepropetrovsk (1975), p. 82–89.

S. B. Vakarchuk, Interpolation by bilinear splines, Math. Notes, 47, № 5, 441–444 (1990). DOI: https://doi.org/10.1007/BF01158084

S. B. Vakarchuk, K. Yu. Myskin, Some problems of simultaneous approximation of functions of two variables and their derivatives by interpolation bilinear splines, Ukr. Math. J., 57, № 2, 173–185 (2005). DOI: https://doi.org/10.1007/s11253-005-0180-0