The Interpolation of Monotone Matrix Functions

Abstract

Let f (x) be a sufficiently smooth function defined on an interval (a, b) of the real axis and let S be a set of N numbers in that interval. We write the elements of S in the form

$${x_1} \leqslant {x_2} \leqslant \cdots \leqslant {x_N}$$

and shall say that a function φ(x) interpolates f(x) at S if φ(x) is a rational function of degree at most N/2 having the property that

$${[{\lambda _1},{\lambda _2}, \cdots ,{\lambda _m}]_f} = {[{\lambda _1},{\lambda _2}, \cdots ,{\lambda _m}]_\varphi }$$

for every subset λ₁, λ₂,…, λ_m of S. Thus the function φ(x) is a solution to the Cauchy interpolation problem associated with the points of S and the function f (x).