Monotone approximation

Three-monotone spline approximation

For r ≥ 3, n ∈ N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differences [x0, x1, x2, x3] f are nonnegative for all choices of distinct points x0, . . . , x3 in [a, b]), we construct a spline s of degree r and of minimal defect (i.e., s ∈ Cr−1[a, b]) with n −1 equidistant knots in (a, b), which is also 3-monotone and satisfies ‖ f − s‖L∞[a,b] ≤ cω...

Monotone trigonometric approximation

Let f ∈ C[−ω, ω], 0 < ω < π, be nonlinear and nondecreasing. We wish to estimate the degree of approximation of f by trigonometric polynomials that are nondecreasing in [−ω, ω]. We obtain estimates involving the second modulus of smoothness of f and show that one in general cannot have estimates with the third modulus of smoothness.

Negative Theorems on Monotone Approximation

Monotone Approximation of Decision Problems

Many decision problems exhibit structural properties in the sense that the objective function is a composition of different component functions that can be identified using empirical data. We consider the approximation of such objective functions, subject to general monotonicity constraints on the component functions. Using a constrained B-spline approximation, we provide a data-driven robust o...

Pointwise estimates for 3-monotone approximation

We prove that for a 3-monotone function F ∈ C[−1, 1], one can achieve the pointwise estimates |F(x) − Ψ(x)| ≤ cω3(F, ρn(x)), x ∈ [−1, 1], where ρn(x) := 1 n2 + √ 1−x2 n and c is an absolute constant, both with Ψ , a 3-monotone quadratic spline on the nth Chebyshev partition, and with Ψ , a 3-monotone polynomial of degree ≤ n. The basis for the construction of these splines and polynomials is th...