Approximation by generalized bivariate Kantorovich sampling type series

The purpose of this paper is to construct a bivariate generalization of new family of Kantorovich type sampling operators $(K_w^{\varphi}f)_{w>0}.$ First, we give the pointwise convergence theorem and a Voronovskaja type theorem for these Kantorovich generalized sampling series. Further, we obtain the degree of approximation by means of modulus of continuity and quantitative version of Voronovskaja type theorem for the family $(K_w^{\varphi}f)_{w>0}.$ Finally, we give some examples of kernels such as box spline kernels and Bochner-Riesz kernel to which the theory can be applied.