On a modified operator associated with  x ^4/3 : asymptotic analysis and improved order of approximation

Abstract

In this work, we introduce a new operator M _n , constructed using a partial differential equation, that is structurally similar to that of the exponential-type operator G _n associated with p ( x ) = x ^4/3 , x ∈ (0, ∞). However, unlike G _n , the operator M _n is not of exponential-type and gives better approximation for various functions in suitable subintervals of (0, ∞). We also provide a concise representation of this operator in terms of special functions. A complete asymptotic expansion for the operator M _n is established. Furthermore, a quantitative Voronovskaya-type estimate is obtained in terms of the weighted modulus of continuity. An approach to further enhance the order of approximation is also proposed. Finally, the effectiveness of the operator M _n is demonstrated through graphical illustrations. A comparison between the true approximation error and truncated asymptotic errors is also provided by using numerical tables.