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

## by Shayne Waldron

## Abstract:

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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