A complete asymptotic expansion for the semi-exponential operators related to $$x^{3}$$

Abstract In the present paper, we study the asymptotic properties of the semi-exponential operator connected with $$p\left( x\right) =x^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mfenced> <mml:mi>x</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> . The main result is a pointwise complete asymptotic expansion valid for locally smooth functions of exponential growth. All coefficients are derived and explicitly given. Furthermore, we characterize classes of functions f for which the semi-exponential operator connected with $$p\left( x\right) =x^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mfenced> <mml:mi>x</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> provides asymptotically a better approximation than the corresponding operator of exponential type. Finally, we present numerical examples which illustrate the better rate of convergence.