Monotone approximation by algebraic polynomials
Authors:
G. G. Lorentz and K. L. Zeller
Journal:
Trans. Amer. Math. Soc. 149 (1970), 1-18
MSC:
Primary 41.40
DOI:
https://doi.org/10.1090/S0002-9947-1970-0285843-5
MathSciNet review:
0285843
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Abstract | References | Similar Articles | Additional Information
Abstract: A given real continuous function f on [a, b] is approximated by polynomials of degree n that are subject to certain restrictions. Let
be given integers,
, given signs. It is assumed that
has the sign of
. Theorems are obtained which describe the polynomials of best approximation, and (for
) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets A, where
attains its maximum, and the sets
with
. Conditions are discussed which these sets must satisfy for a polynomial
of best approximation for f. Numbers of the points of sets A,
are studied, the possibility of certain extreme situations established. For example, if
, it is possible that
.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1970-0285843-5
Keywords:
Monotone approximation,
polynomials of best approximation,
admissible sets,
minimal polynomials,
basic polynomials of Lagrange interpolation,
Birkhoff interpolation,
free incidence matrices,
poised incidence matrices
Article copyright:
© Copyright 1970
American Mathematical Society