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<i>Let φ(·) be a nonconstant, bounded, and monotome-increasing continuous function. Let ''I''<sub>''m''<sub>0</sub></sub> denote the ''m''<sub>0</sub>-dimensional unit hypercube [0,1]<sup>''m''<sub>0</sub></sup>. The space of continuous functions on ''I''<sub>''m''<sub>0</sub></sub> is denoted by ''C''(''I''<sub>''m''<sub>0</sub></sub>). Then, given any function ''f'' Э ''C''(''I''<sub>''m''<sub>0</sub></sub>) and є > 0, there exist an integer ''m''<sub>1</sub> and sets of real constants ''α''<sub>''i''</sub>, ''b''<sub>''i''</sub> and ''w''<sub>''ij''</sub>, where ''i'' = 1, ..., ''m''<sub>1</sub> and ''j'' = 1, ..., ''m''<sub>0</sub> such that we may define:</i>
: <math>▼
▲<math>
▲ F( x_1 , ..., x_{m_0} ) =
</math>
<math>
\sum_{i=1}^{m_1} \Alpha_i \varphi \left( \sum_{j=1}^{m_0} w_{i,j} x_j + b_i\right)
</math>
<i>as an approximate realization of the function ''f''; that is,</i></div>
: <math>▼
▲<math>
▲ | F( x_1 , ..., x_{m_0} ) - f ( x_1 , ..., x_{m_0} ) | < \epsilon
</math>
<i>
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