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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 0285843
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Abstract: A given real continuous function f on [a, b] is approximated by polynomials $ {P_n}$ of degree n that are subject to certain restrictions. Let $ 1 \leqq {k_1} < \cdots < {k_p} \leqq n$ be given integers, $ {\varepsilon _i} = \pm 1$, given signs. It is assumed that $ P_n^{({k_i})}(x)$ has the sign of $ {\varepsilon _i},i = 1, \ldots ,p,a \leqq x \leqq b$. Theorems are obtained which describe the polynomials of best approximation, and (for $ p = 1$) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets A, where $ \vert f(x) - {P_n}(x)\vert$ attains its maximum, and the sets $ {B_i}$ with $ P_n^{({k_i})}(x) = 0$. Conditions are discussed which these sets must satisfy for a polynomial $ {P_n}$ of best approximation for f. Numbers of the points of sets A, $ {B_i}$ are studied, the possibility of certain extreme situations established. For example, if $ p = 1,{k_1} = 1,n = 2q + 1$, it is possible that $ \vert A\vert = 3,\vert B\vert = n$.

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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