On Bernstein's comparison theorem, Peano kernels of constant sign and near-minimax approximation

by Shayne Waldron


Some basic properties of what are called `B(ernstein)-monotone' seminorms are investigated. These lie between the classes of monotone and sign-monotone seminorms. It is seen that these seminorms arise naturally in Bernstein's comparison theorem, the description of Peano kernels of constant sign, and in near-minimax approximations. A number of new results are obtained including some sufficient conditions for a projection to be near-minimax which are easily seen to be satisfied by all the known examples, and a characterisation of the Peano kernels of constant sign where derivatives are replaced by divided differences.

Keywords: monotone seminorm, B-monotone seminorm, sign-monotone seminorm, property B of order k, Bernstein's comparison theorem, Peano kernel, divided difference, Hermite interpolation, Steffensen's differentiation formula, positive linear operator, Bernstein operator, variation diminishing splines, near-minimax approximation, Chebyshev polynomial

Math Review Classification: 41A10, 41A65, 41A80 (primary), 41A05, 41A55, 46E99, 65J05 (secondary)

Length: 17 pages

Comment: Written in TeX

Last updated: 12 May 1997

Status: Submitted


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