Riemann–Liouville Fractional Integral Type Deep Neural Network Kantorovich Operators

This paper introduces a novel family of Kantorovich-type deep neural network operators based on Riemann–Liouville fractional integrals. Building upon the work of Costarelli (Math Model Anal 27(4):547–560, 2022) and Sharma and Singh (J Math Anal Appl 533(2):128009, 2024), we investigate the approximation properties of these operators in the spaces \({{\mathcal {C}}}({\mathscr {I}})\) $\mathcal{C}(\mathcal{I})$ (the space of all continuous functions on \({\mathscr {I}}:=[-1,1]\) $\mathcal{I}:=[-1,1]$ ) and \({\mathscr {L}}_{{\mathcalligra {p}}}({\mathscr {I}})\) ${\mathcal{L}}_{\mathcal{p}}(\mathcal{I})$ (the space of all \({\mathcalligra {p}}\) $\mathcal{p}$ -th Lebesgue integrable functions on \({\mathscr {I}}\) $\mathcal{I}$ , \(1\le {\mathcalligra {p}}<\infty\) $1\le \mathcal{p}<\infty$ ). We establish point-wise and uniform convergence results for both single and multi-hidden layer networks in the spaces \({{\mathcal {C}}}({\mathscr {I}})\) $\mathcal{C}(\mathcal{I})$ and \({\mathscr {L}}_{{\mathcalligra {p}}}({\mathscr {I}})\) ${\mathcal{L}}_{\mathcal{p}}(\mathcal{I})$ , \(1\le {\mathcalligra {p}}<\infty\) $1\le \mathcal{p}<\infty$ . Our analysis leverages auxiliary approximation results for the single-hidden layer case to derive density theorems for the two-hidden layer and multi-hidden layer scenarios. Finally, we discuss specific examples of sigmoidal activation functions that are compatible with our proposed operators.