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We give direct and converse results for the weighted approximation of functions with endpoint or inner singularities by Bernstein operators.

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Weighted approximation of functions by Szász--Mirakyan-type operators

Sz\'asz--Mirakyan operator, weighted modulus of smoothness, direct and converse results

Acta Mathematica Hungarica
Authors: Biancamaria Della Vecchia, Giuseppe Mastroianni, and Szabados J


We give error estimates for the weighted approximation of functions with singularities at the endpoints on the semiaxis by some modifications of Sz\'asz--Mirakyan operators. To do so, we define a new weighted modulus of smoothness and prove its equivalence to the weighted K-functional. Also, the class of functions for which the modified Sz\'asz--Mirakyan operator can be defined will be extended to a much wider set than for the original operator.

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