## Abstract

Let I be a finite interval and r, s ∈ ℕ. Given a set M, of functions defined on I, denote by Δ_{+}^{s}M the subset of all functions y ∈ M such that the s-difference Δ _{τ}^{s}y(·) is nonnegative on I, ∀τ > 0. Further, denote by Δ_{+}^{s}W_{p}^{r}, the class of functions x on I with the seminorm ∥x ^{(r)}∥_{Lp} ≤ 1, such that Δ_{τ} ^{s}x ≥ 0, τ > 0. Let M_{n}(h_{k}):= {∑_{i=1}^{n}c_{i}h_{k}(w_{i}t- θ_{i}) | c_{i}, w_{i}, θ_{i} ∈ ℝ}, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions h_{k}(t) = t_{+} ^{k}, t ∈ ℝ, k ∈ ℕ_{0}. We give two-sided estimates both of the best unconstrained approximation E(Δ _{+}^{s}W_{p}^{r},M_{n}(h _{k}))_{Lq}, k = r - 1, r, s = 0, 1,...,r + 1, and of the best s-monotonicity preserving approximation E(Δ_{+}^{s}W _{p}^{r},Δ_{+}^{s}M_{n}(h _{k}))_{Lq}, k = r - 1, r, s = 0, 1,...,r + 1. The most significant results are contained in theorem 2.2.

Original language | English |
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Pages (from-to) | 347-366 |

Number of pages | 20 |

Journal | Advances in Computational Mathematics |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - May 2004 |

## Keywords

- Free-knot spline
- Order of approximation
- Relative width
- Shape preserving
- Single hidden layer perceptron model