Free-knot splines approximation of s-monotone functions

V. N. Konovalov*, D. Leviatan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let I be a finite interval and r, s ∈ ℕ. Given a set M, of functions defined on I, denote by Δ+sM the subset of all functions y ∈ M such that the s-difference Δ τsy(·) is nonnegative on I, ∀τ > 0. Further, denote by Δ+sWpr, the class of functions x on I with the seminorm ∥x (r)Lp ≤ 1, such that Δτ sx ≥ 0, τ > 0. Let Mn(hk):= {∑i=1ncihk(wit- θi) | ci, wi, θi ∈ ℝ}, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions hk(t) = t+ k, t ∈ ℝ, k ∈ ℕ0. We give two-sided estimates both of the best unconstrained approximation E(Δ +sWpr,Mn(h k))Lq, k = r - 1, r, s = 0, 1,...,r + 1, and of the best s-monotonicity preserving approximation E(Δ+sW pr+sMn(h k))Lq, k = r - 1, r, s = 0, 1,...,r + 1. The most significant results are contained in theorem 2.2.

Original languageEnglish
Pages (from-to)347-366
Number of pages20
JournalAdvances in Computational Mathematics
Issue number4
StatePublished - May 2004


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


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