Three-monotone spline approximation

Abstract

For r ≥ 3, n ∈ N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differenceswe construct a spline s of degree r and of minimal defect (i.e., s ∈ C r −1 [a, b]) with n − 1 equidistant knots in (a, b), which is also 3-monotone and satisfieswhere ω 4 ( f, t, [a, b]) ∞ is the (usual) fourth modulus of smoothness of f in the uniform norm. This answers in the affirmative the question raised in [8, Remark 3], which was the only remaining unproved Jackson-type estimate for uniform 3-monotone approximation by piecewise polynomial functions (ppfs) with uniformly spaced fixed knots.Moreover, we also prove a similar estimate in terms of the Ditzian-Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the L p norm with p < ∞. At the same time, positive results in the L p case with p < ∞ are still valid if one allows the knots of the approximating ppf to depend on f while still being controlled.These results confirm that 3-monotone approximation is the transition case between monotone and convex approximation (where most of the results are "positive") and k-monotone approximation with k ≥ 4 (where just about everything is "negative").

Determination of a mathematical model of the thin-film photovoltaic panel (CIS) based on measurement data

“…Mathematical models were determined using multivariate polynomial approximation (many variables) [4]; the detailed examples of calculation shown later in the paper relate to Shell ST20 photovoltaic panel [14].…”