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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, C([c,d])$$ C\left(\left[c,d\right]\right) $$ and Lp([c,d]),1≤p<∞$$ {\mathtt{L}}&amp;#x0005E;{\mathtt{p}}\left(\left[c,d\right]\right),1\le \mathtt{p}&lt;\infty $$, 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 Lp,1≤p<∞$$ {\mathtt{L}}&amp;#x0005E;{\mathtt{p}},1\le \mathtt{p}&lt;\infty $$, spaces. [ABSTRACT FROM AUTHOR]

The paper deals with the approximation of first order derivative of a function by the first order derivative of Szász-type operators based on Charlier polynomials introduced by Varma and Taşdelen [ 20 ]. The uniform convergence theorem, Voronovskaja type asymptotic theorem and an estimate of error in terms of the second order modulus of continuity of the derivative of the function are investigated. Further, it is shown that linear combinations of the derivative of the above operators converge to the derivative of function at a faster rate. Finally, an estimate of error in the approximation is obtained in terms of the (2 k + 2) t h (2 k + 2) t h order modulus of continuity using Steklov mean. [ABSTRACT FROM AUTHOR]