Monotone approximation by algebraic polynomials

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 $|f(x) - {P_n}(x)|$ 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 $|A| = 3,|B| = n$.