Multivariate φ-Variational Approximation of Mellin-Type Nonlinear Integral Operators via Summability Methods

In this paper, we construct the nonlinear form of multidimensional Mellin-type integral operators and improve them using summability methods. We utilize Tonelli sense convergence in φ-variation, incorporating the Haar measure into our framework. Additionally, we study the rate of approximation and provide a characterization theorem for functions that are absolutely continuous in the Tonelli sense. Finally, we present illustrations of our approximations to support the theoretical findings.