Neural network Kantorovich operators activated by smooth ramp functions

In the present article, we introduce a Kantorovich variant of the neural network interpolation operators activated by smooth ramp functions proposed by Qian and Yu (2022). We discuss the convergence of these operators in the spaces, and , and establish some direct approximation theorems. Further, we derive the converse results by means of Berens–Lorentz lemma and Peetre's K‐functional. We present a multivariate version of the aforementioned Kantorovich neural network interpolation operators and investigate the direct and converse results in the continuous and , spaces.