Nonexistence of best alternating approximations on subsets

Abstract

The existence of best Chebyshev approximations by an alternating family on closed subsets of an interval is considered. In varisolvent approximation, existence on subsets of sufficiently low density is guaranteed if the best approximation on the interval is of maximum degree. The paper studies the case in which the best approximation is not of maximum degree and shows that in many common cases, no such guarantee is possible. Let Y be a compact subset of [a,,/]] and C(Y) be the space of continuous functions on Y. For g E C(Y) define llglly = sup{lg(x)A: x E Y}, lIlgH = 110[al. Let F be an approximating function on [a,,8/] with parameter space P. The Chebyshev problem on Y is, given f E C(Y), to find a parameter A * to minimize lif F(A, .)IIy over A E P. Such a parameter A* is called best and F(A*, ) is called best to f on Y. We will consider only approximating functions for which best approximations on [a,,8]1 are characterized by alternation [10, pp. 17-21], [6], which implies uniqueness of best approximations. Following Cheney, we define the density of Y in [a,,8]1 to be IYI = max{inf{x -yl: y Y} a 0, there is a parameter B E P such that IF(B, a) -y < e and Received by the editors October 2, 1975. AMS (MOS) subject classifications (1970). Primary 41A50. Copyright ? 197'. Aniwrican Mathematical Society