The degree of piecewise monotone interpolation

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.