The Interpolation of Monotone Matrix Functions

  • William F. DonoghueJr.
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 207)


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,…, λm 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).


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Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • William F. DonoghueJr.
    • 1
  1. 1.University of CaliforniaIrvineUSA

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