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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The degree of piecewise monotone interpolation

Authors: Eli Passow and Louis Raymon
Journal: Proc. Amer. Math. Soc. 48 (1975), 409-412
MSC: Primary 41A15
MathSciNet review: 0430608
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Abstract: Let $ 0 = {x_0} < {x_1} < \cdots < {x_k} = 1$ and let $ {y_0},{y_1}, \cdots ,{y_k}$ be real numbers such that $ {y_{j - 1}} \ne {y_j},j = 1,2, \cdots ,k$. Estimates are obtained on the degree of an algebraic polynomial $ p(x)$ that interpolates the given data piecewise monotonely; i.e., such that (i) $ p({x_j}) = {y_j},j = 0,1, \cdots ,k$, and such that (ii) $ p(x)$ is increasing on $ {I_j} = ({x_{j - 1}},{x_j}{\text{) if }}{y_j} < {y_{j - 1}}$, and decreasing on $ {I_j}$. if $ {y_j} < {y_{j - 1}},j = 1,2, \cdots ,k$. The problem is seen to be related to the problem of monotone approximation.

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Keywords: Monotone interpolation, monotone approximation, comonotone approximation
Article copyright: © Copyright 1975 American Mathematical Society