Weighted approximation by Bernstein quasiinterpolants for functions with singularities

Abstract

We introduce a new type of modified Bernstein quasi-interpolants, which can be used to approximate functions with singularities. We establish direct, inverse, and equivalent theorems of the weighted approximation of this modified quasi-interpolants. Some classical results on approximation of continuous functions are generalized to the weighted approximation of functions with singularities.

Keywords

Quasi-interpolants / function with singularities / Bernstein operator / weighted approximation / equivalent theorem

References

[1] Della Vechhia D, Mastroianni G. Szabados J. Weighted approximation of functions with endpoint and inner singularities by Bernstein operators. Acta Math Hungar, 2004, 103: 19-41

[2] Diallo A T. Rate of convergence of Bernstein quasi-interpolants, 1995

[3] Ditzian Z, Totik V. Moduli of smoothness, 1987, Berlin, New York: Springer-Verlag

[4] Guo S S, Liu G F. Strong converse inequality for left Bernstein-Kantorovich quasiinterpolants. Acta Math Sinica (China Ser), 2010, 53: 107-116

[5] Lorentz G G. The degree of approximation by polynomials with positive coefficients. Math Ann, 1963, 151: 239-251

[6] Mache P, Mache D H. Approximation by Bernstein quasi-interpolants. Numer Funct Anal Optimiz, 2001, 22: 159-175

[7] Salonniére P. A family of Bernstein quasi-interpolants on [0, 1]. Approx Theor Appl, 1992, 8: 62-76

[8] Yu D S. Weighted approximation of functions with singularities by combinations of Bernstein operators. Appl Math Comput, 2008, 206: 906-918

[9] Zhang G S. Approximation properties of quasi-interpolants of some famous linear operators, 2004

[10] Zhou D X. On smoothness characterized by Bernstein operators. J Approx Theory, 1994, 81: 303-315