A $C^{\infty}$ rational quasi-interpolation operator for functions with jumps without the Gibbs phenomenon

The study of quasi-interpolation has gained significant importance in numerical analysis and approximation theory due to its versatile applications in scientific and engineering fields. This technique provides a flexible and efficient alternative to traditional interpolation methods by approximating data points without requiring the approximated function to pass exactly through them. This approach is particularly valuable for handling jump discontinuities, where classical interpolation methods often fail due to the Gibbs phenomenon. These discontinuities are common in practical scenarios such as signal processing and computational physics. In this paper, we present a $C^{\infty}$ rational quasi-interpolation operator designed to effectively approximate functions with jump discontinuities while minimizing the issues typically associated with traditional interpolation methods.