Almost monotone approximation in L∞

Ubiquitous monotone and almost-monotone Boolean functions in systems biology

Systems Biology is quickly becoming a major focus of applied and computational mathematics research. While traditionally numerics and continuous modeling techniques have been used, in recent years it has become apparent that qualitative modeling, data quantization and data mining questions play a central role for the field. Monotone Boolean functions are ubiquitous in these models, and, surpris...

Monotone partitions and almost partitions

Article history: Received 28 April 2014 Received in revised form 14 October 2014 Accepted 14 October 2014 Available online 31 October 2014 MSC: 54D20

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