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Almost everywhere approximation capabilities of double Mellin approximate identity neural networks

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Abstract

The research works in approximation theory of artificial neural networks is still far from completion. To fill a gap in this issue, this study focuses on the almost everywhere approximation capabilities of single-hidden-layer feedforward double Mellin approximate identity neural networks. First, the notion of double Mellin approximate identity is introduced. Using this notion, an auxiliary theorem is proved. The auxiliary theorem provides a connection between a class of double Mellin convolution linear operators and the notion of almost everywhere convergence. This theorem is applied to prove a main theorem. The proof of the main theorem is based on the notion of epsilon-net. The main theorem shows almost everywhere approximation capability of single-hidden-layer feedforward double Mellin approximate identity neural networks in the space of almost everywhere continuous bivariate functions on \( \mathbb {R}_{+}^{2} \). Moreover, similar results are obtained in the spaces of almost everywhere Lebesgue integrable bivariate functions on \( \mathbb {R}_{+}^{2} \).

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Acknowledgments

We would like to thank Dr. Aniello Castiglione managing editor Soft Computing for his encouragement. We are also thankful to the anonymous reviewers for their valuable suggestions which have improved the paper. The first author was supported by a postdoctoral research fellowship program at Universiti Sains Malaysia.

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Correspondence to Saeed Panahian Fard.

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Communicated by V. Loia.

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Fard, S.P., Zainuddin, Z. Almost everywhere approximation capabilities of double Mellin approximate identity neural networks. Soft Comput 20, 4439–4447 (2016). https://doi.org/10.1007/s00500-015-1753-y

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