Nonlinear approximation in N-dimension with the help of summability methods

In this paper, we approximate to functions in N-dimension by means of nonlinear integral operators of the convolution type. Our approximation is based on not only the uniform norm but also the variation semi-norm in Tonelli's sense. We also study the rates of convergence. To get more general results we mainly use regular summability methods in the approximation. We construct some significant applications including the Cesaro approximation, the almost approximation, the rates of convergence based on certain summability methods. Furthermore, we display some graphical illustrations verifying the approximation and evaluate numerical computations giving approximation errors.