On computational algorithms for real-valued continuous functions of several variables

Area of Science:

• Computational mathematics
• Function approximation
• Algorithm development

Background:

• Kolmogorov's dimension-reducing superpositions (1957) provided early prototypes for function representation.
• Hecht-Nielsen (1987) linked these superpositions to feedforward neural networks, increasing interest.
• Original superpositions used devil's staircase functions, limiting their computational utility.

Purpose of the Study:

• To investigate algorithms for computing superpositions of real-valued continuous functions.
• To explore the application of space-filling curves in function approximation.
• To present novel algorithms for two-variable function superposition.

Main Methods:

• Analysis of algorithms based on space-filling curves.
• Examination of the relationship between Kolmogorov's superpositions and computational algorithms.
• Development of two specific algorithms for two-variable function superposition, including one using the Hilbert curve.

Main Results:

• Identified limitations in early algorithms due to space-filling curve characteristics.
• Presented two new algorithms for computing superpositions of two-variable functions.
• Demonstrated the utility of specific space-filling curves, like the Hilbert curve, in these algorithms.

Conclusions:

• Space-filling curves offer a basis for developing algorithms for function superposition.
• The developed algorithms provide efficient methods for approximating two-variable continuous functions.
• This work advances the computational application of dimension-reducing superpositions.