Tchebycheff Systems and Total Positivity

This chapter is not concerned with n-widths but is a discussion of Tchebycheff (T-) systems (often written Chebyshev) and total positivity which are important tools in the exact determination of n-widths and in the identification of optimal subspaces for many of the examples considered in this work. While we do assume that the reader is familiar with some of the basic results of approximation theory and related matters, we cannot assume that the reader also has a familiarity with T-systems and total positivity. It transpires that the theory of T-systems and total positivity is intimately connected with the problem of zero counting and oscillations of functions and, as such, is basic to the study of L∞ and L1 approximations, as shall be evident from results of succeeding chapters. Perhaps surprisingly it is also important in the L2 theory of n-widths.