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An estimation of the approximation rate by positive linear operators of functions defined on the positive half line that have finite limit at infinity is discussed. For the application of the obtained results, we introduce the positive linear operators which preserve2a⁢x2^{ax},a&gt;0a&gt;0, and obtain their uniform convergence, order of approximation via a certain weighted modulus of continuity, and a quantitative Voronovskaya type theorem.

The rate of approximation of functions in an infinite interval by positive linear operators