Hyperbolic sigma-pi neural network operators for compactly supported continuous functions (Q1923886)

Let \(f\in C(\mathbb{R}^n)\) be a \(\Delta\)-Lipschitz continuous function with \(f(x)= 0\) for \(x\not\in (- 1,1)^n\) and let \(\sigma\) be a sigmoidal function, i.e. \(\sigma: \mathbb{R}\to \mathbb{R}\) is bounded, \(\sigma(\xi)\to 0\) as \(\xi\to -\infty\) and \(\sigma(\xi)\to 10\) as \(\xi\to \infty\). There is defined a sequence of linear hyperbolic-type operators \(\Delta^{(1/N)}\) \((N= 1, 2,\dots)\), \(\Delta^{(1/N)}(f):\mathbb{R}^n\to\mathbb{R}\) based on a sigmoidal function \(\sigma\) and it is proved that \(\Delta^{(1/N)} f(x)\to f(x)\) as \(N\to \infty\) uniformly for \(x\in [- 1, 1]^n\). These operators are interpreted as concrete real-time realizations of three-layer feedforward neural networks with sigma-pi units in the hidden layer.